\documentclass[12pt,epsf]{article}
        \usepackage{graphicx}
\usepackage{amsmath}                                                              
          \usepackage{amssymb}

    

\parskip 10pt
\input{tcilatex}

\begin{document}


\baselineskip 16pt

\parindent 24pt

\parskip 10pt

\titlepage
                   


\begin{center}
\begin{LARGE}
 
 \textbf{ A Theory of Rivalry: Does Number Two Try Harder? }
\\[0pt]
 

\end{LARGE}

 

April 29, 1989 \\[0pt]

 

Eric Rasmusen \\[0pt]

 

\textit{Abstract}
\end{center}

   Rivalry occurs when one player exerts effort to improve or maintain
his standing relative to another player. In the model of this paper,
a player can be either behind, even with, or ahead of his rival, and
effort stochastically improves his position.  In a one-period game,
both players exert the same effort, exerting more if they are even
than otherwise.  In all but the last period of a $T$-period game, the
player that is ahead exerts more effort, and in any period both
players exert more if they are even than if one is ahead.
Applications to innovation, elections, wars, arms races, and
advertising are suggested.
 
  I abandoned this paper because I thought that too many other similar papers were being published, and mine would not add much value. 

\begin{small}

\noindent
  \hspace*{20pt} Indiana University Foundation Professor,
Department  of Business Economics and Public Policy,
  Kelley School of
Business,Indiana University,  BU 456, 1309 E. 10th Street,
Bloomington, Indiana, 47405-1701.
Office: (812) 855-9219. Fax: 812-855-3354.
Erasmuse@indiana.edu.\newline
Php.indiana.edu/$\sim$erasmuse.  
   

\noindent
   I   thank Sushil Bikhchandani, David Hirshleifer, Ivan
Png, Steven Lippman, George Tsebelis, and Emmanuel Petrakis for
helpful comments. 
 

 
 
  Draft: 22.2.\\
  UCLA AGSM Business Economics Working Paper \#88-1.\\ 
  Former titles: ``Investment Under  Technical Change,'' ``Races for
 Relative Position, Ending and  Endless, ``Rivalry in a Dynamic
 Innovation Game.''  


   \end{small}

%---------------------------------------------------------------
        \newpage
\noindent
{\bf 1. INTRODUCTION.}

  Avis rent a car's motto was ------.  Is it true that a follower
tries harder than a leader--- what we might call the Avis Question?
Intuition suggests several plausible, but incompatible, answers: the
follower gets discouraged and slows down; the leader becomes smug and
slows down; the lead does not make any difference.  We will see that
the lead does make a difference: in the model below, the leader
exerts more effort than the follower, but both exert more when they
are even than when one is ahead.  

 The concept of rivalry applies to a variety of situations.  Will the
country that is winning a war fight harder? If it is ahead in an arms
race will it try hard to maintain its lead? Will the firm with the
biggest share of the market indulge in the most advertising?  Will
the firm with the best product spend the most on research? Will the
currently dominant bird most energetically fight for rank in the
pecking order?  Will the faction that currently has the president's
ear exert the most effort to keep his favor? I will return to these
questions towards the end of the paper, and suggest arguments
applicable to all these situations to varying degrees.

  The formal model laid out in the next section is a $T$-period game
between two players contending for shares of a fixed prize that can
depend both on the current state and the final state in period $T$.
In each period the players simultaneously choose effort levels that
stochastically affect the state the next period, which can take three
values: either of the two players being ahead, or both being even
with each other.  or behind his rival in the next period.  The model
will be described in general terms for most of the paper, but readers
may find it helpful to think of a dynamic innovation model in which
two firms choose research spending each year in a battle over shares
of a fixed market.

 Some of the assumptions are special--- the restriction to two rivals
and three states, for example--- but I hope the model will confirm
one set of arguments for why the leader might exert more effort. A
number of papers have addressed the Avis Question and come up with
various answers. Following the terminology of Vickers (1986), the
answer could be either Increasing Dominance (the leader tries harder,
and tends to remain the leader) or Action-Reaction (the follower
tries harder, and the lead tends to alternate). The papers addressing
the question will be discussed below in Section 5.  One overall
lesson, however, is that general models are hard to construct and a
great many influences are at work. Some papers confine themselves to
a few states; others use a special assumption on how effort
translates to achievement.  Merely to answer the Avis Question for a
particular model is useless; one ought to use the particular model to
simply and convincingly illustrate a particular effect of rivalry.

\noindent
 {\bf Plan of the Paper.}\\
 1. Introduction.\\
 2. Description of the model.\\
 3. The game with one period.\\
 4. The game with two or more periods.\\
 5. Relaxing assumptions.\\
 6. The rivalry literature.\\
 7. Applications.


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

\newpage
\noindent
  {\bf 2. THE MODEL.}

  Two risk-neutral players, Alpha and Beta, are playing a $T$-period
game of complete information with simultaneous moves and uncertainty.
Let us adopt the point of view of Alpha.  An achievement level is
associated with each player, and the important state variable is
Alpha's achievement lead over Beta.  The lead can take three values:
Alpha can be ahead ($\alpha$), even (0), or behind ($\beta$).  Let us
denote the expected value of Alpha's payoff starting from the
beginning of period $t$ by $V_t^{\alpha}, V_t^0$, or $V_t^\beta$,
depending on the lead.

  Alpha and Beta choose efforts $a$ and $b$ simultaneously each
period. Alpha's lead rises with probability $f(a,b)$, falls with
probability $g(a,b)$, and remains unchanged with probability $1 - f -
g$.  Subscripts and superscripts attached to these functions will
denote the state of the game: the probability of an advance in period
$t$ when Alpha is ahead is $f_t^{\alpha}$, for example.  If Alpha is
ahead and chance would call for the lead to increase, Alpha merely
stays ahead, so any effort by Alpha in that state is intended purely
to block Beta.

  Unlike most models of innovation under uncertainty, this model does
not specify a functional form for the function $f$.  Instead, let us
make general assumptions about the properties of $f$, an approach
that will require more explanation in setting up the model, but which
will point out which properties are important. The assumptions have
two objectives: to guarantee existence of equilibrium efforts, and to
isolate the incentive effects of the lead by ruling out purely
technological advantages for the leader or the follower.

  Since $f$ is a probability, it takes values on the interval
$[0,1]$.  Let us assume that $f$ does not depend on the state, except
through the choices of $a$ and $b$. Let us also assume that $f$ is
concave and increasing in Alpha's effort, and decreasing and convex
in Beta's effort, so that Alpha's effort tends to increase the lead,
but with diminishing returns, while Beta's effort tends to prevent
that increase, also with diminishing returns.  These assumptions
imply that
 $$ 
\begin{array}{lr}
 \frac{\partial f}{\partial a} > 0 & (1a)\\
 \frac{\partial ^2f}{\partial  a^2} < 0 & (1b)\\
 \frac{\partial f}{\partial  b} < 0 & (1c)\\
 \frac{\partial ^2f}{\partial  b^2} > 0  & (1d)\\
 \end{array}
 $$
 \setcounter{equation}{1}

 Two further assumptions will help avoid extreme equilibria.  First,
assume that as Alpha's effort goes to zero, his effort's marginal
product rises to infinity:
  \begin{equation} \label{e2}
 \stackrel{Limit}{a \rightarrow 0} \frac{\partial f}{\partial a} =
\infty. 
 \end{equation}
  Assumption (\ref{e2}) guarantees an interior solution to the
player's maximization problem, so that both players exert positive
effort in equilibrium, whatever the state.$^1$ Note that since the
players will always exert effort in equilibrium, whether or not
changes in the lead would occur spontaneously when both players exert
zero effort is unimportant to the model.

  Second, assume that direct effects are stronger than indirect
effects: when Alpha increases his effort, the effect on Alpha's own
marginal product of effort is greater than on Beta's.  This includes
the case in which each player's marginal product of effort is
independent of the other player's effort, a case which is perhaps the
most central for this analysis, since it rules out a purely
technological effect of the size of one player's effort on the
other's effort.
  \begin{equation} \label{e3}
 \frac{ \partial^2f}{\partial a \partial b} \geq 0.
 \end{equation}

 Let us make two additional assumptions to remove the possibility of
purely technological differences between the leader and the follower
or Alpha and Beta.  First, assume that the function $f$ is symmetric
in $a$ and $b$, so that effort does not have different effects on
``offense'' and ``defense.''  Such an asymmetry would tilt the
results in obvious ways.  If, for example, effort were useless in
defense ($\frac{\partial f}{\partial b} = 0$), one would not expect a
player who was ahead to exert any effort. Let us therefore assume
that
  \begin{equation} \label{e4}
 \frac{\partial f}{\partial a} = -\frac{\partial f}{\partial b} \;
{\rm if} \; a=b. 
 \end{equation}

  Second, assume that the two players are identical except for
differences in who is ahead.  This implies that the form of the
functions by which they advance is the same; i.e., 
  \begin{equation} \label{e5}
 f(x,y) = g(y,x) \; \forall x,y,
\end{equation}
 which implies, using our earlier assumptions, that $ \frac{\partial
g}{\partial b} > 0, \frac{ \partial^2g}{\partial b^2} < 0,
\frac{\partial g}{\partial a} < 0$, and $\frac{ \partial^2g}{\partial
a^2} > 0.$

  It remains to specify the payoffs as functions of the state.
Player Alpha's current benefit per period is denoted by $R^{\alpha},
R^0$, or $R^\beta$.  Assume that the total benefit each period is
constant at $\bar{R}$, so that if Alpha is ahead, for example, Beta's
current benefit is $\bar{R} -R^{\alpha}$.  Since the players are
identical, $ R^0 = \bar{R}/2$, from which it follows that
$R^{\alpha}-R^0=R^0-R^\beta$ (from $R^0- R^\beta = R^0 - (\bar{R}-
R^{\alpha}) = R^0 - 2R^0 + R^{\alpha} = R^{\alpha} - R^0$.)
 The assumption of constant total
benefit isolates the effect of rivalry, and excludes consideration of
the kind of effort a lone player might make. The game is zero-sum on
the benefit side, though not on the cost side.  This is a simplifying
assumption; adding socially productive effort would not change the
model significantly.\footnote{xxx Lemma 3 may require removing the
balloon benfit. I hope not.}

   At the end of the game there is a final balloon benefit of
$W^{\alpha}, W^0$, or $W^\beta$ to Alpha, and a balloon benefit of
$\bar{W}$ less Alpha's benefit for Beta.  The amount $W^{\alpha}$, for
example, represents the value of being the permanent leader when the
lead freezes after period $T$.  It would take the value $\frac{\delta
R^{\alpha}}{1-\delta}$ if the flow of benefits continues except for the
absence of changes in the lead, and it would take the value
$R^{\alpha}$ if the benefits simply ended.  Because the firms are
identical, $W^0 = \bar{W}/2$, and $W^{\alpha}-W^0=W^0-W^\beta$. 

 The model includes as special cases: (i) No balloon benefit
($\overline{W}=0$); (ii) The only benefit being the balloon benefit (
$\overline{R}=0$); and (iii) winner-take-all benefits ( $R^{\alpha} =
\overline{R}$ and/or $W^{\alpha}= \overline{W}$).\footnote{xxx NOt
written parallel right now.}

 Let us denote the discount factor by $\delta>0$, which can be either
less than or equal to one, and assume that utility is separable in
benefit and effort, linear in effort, and with a cost of effort
normalized to unity. Linearity is purely for simplicity, and the
analysis would be much the same if the marginal disutility of effort
were increasing.

   Alpha's value function, representing his discounted equilibrium
expected payoff at the beginning of the period, is one of three
equations depending on the lead. Each equation is made up of the
current benefit, the current cost of effort, and the discounted
expected value at the end of the period. If Alpha is ahead at the
start of period $t$, for example, he knows that he will lose the
lead with probability $g$ and keep it with probability $1-g$, so his
value function is
   \begin{equation} \label{e7} 
 V_t^{\alpha} = R^{\alpha} - a_t^{\alpha} + \delta g_t^{\alpha} V_{t+1}^0 +
\delta \left( 1- g_t^{\alpha} \right)V_{t+1}^{\alpha}.
 \end{equation}


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

\bigskip
\noindent
 {\bf The Equilibrium Concept.} 

   As is standard in dynamic games of complete information, an
equilibrium will be defined here as a subgame perfect strategy
combination; i.e., the equilibrium strategies for the entire game
must be Nash strategies for each subgame. Perfectness rules out
non-credible threats such as Alpha threatening to exert very high
effort for the rest of the game if Beta ever exerts positive effort.

  Let us also restrict our attention to symmetric equilibria,
requiring that the equilibrium strategies be the same for the two
players.  This does not mean that the players take the same actions
in a given period. If Alpha is ahead in period $t$, for example, he
might not exert the same effort as Beta, but he does exert the same
effort that Beta would were Beta ahead in period $t$.

 Symmetry of the equilibrium strategies will require some care in the
proofs. We cannot impose the equations in (\ref{e10}) before finding
the first order conditions, because symmetry of the strategies
applies to the equilibrium values of the strategies, not just to any
values.


 For the equilibrium values of the strategies, symmetry requires that
 \begin{equation} \label{e10}
  a_t^0 = b_t^0, 
  a_t^{\alpha} = b_t^\beta, \;\;{\rm and}\; a_t^\beta = b_t^{\alpha}. 
  \end{equation}
  The symmetry assumption is equivalent to finding all the subgame
perfect equilibria and then discarding the asymmetric ones. Its
purpose is to try to exclude equilibria whose properties have more to
do with the expectations of players about each others' strategies
than with fundamentals such as payoffs and initial positions.  It may
be that asymmetric equilibria do exist, although I have no evidence
that they do.




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

\newpage
\noindent
 {\bf THE GAME WITH ONE PERIOD.}

 The time subscript will be dropped in this section since there is
only one period.  Remember, however, that the one-period game is
identical to the last subgame of the two-period or $T$-period game, a
fact that will be useful later.

\noindent
 The value function of equation (\ref{e7}) becomes
  \begin{equation} \label{e12a} 
 V^{\alpha} = R^{\alpha} - a^{\alpha} + \delta g^{\alpha} W^0 + \delta
\left( 1- g^{\alpha} \right)W^{\alpha}, 
 \end{equation}
 and the value functions in the other two states are
\begin{equation} \label{e12b} 
 V^\beta = R^\beta - a^{\beta} + \delta f^\beta W^0 + \delta \left(
1-f^\beta \right)W^\beta
 \end{equation}
 and
\begin{equation} \label{e13} 
 V^0 = R^0 - a^0 + \delta f^0 W^{\alpha} + \delta g^0 W^\beta + \delta
\left( 1- f^0 -g^0 \right) W^0.
 \end{equation}
 Since $ W^0- W^\beta = W^{\alpha} - W^0$,  equations
(\ref{e12a}) through (\ref{e13}) can be rewritten as
  \begin{equation} \label{e14} 
 V^{\alpha} = R^{\alpha} - a^{\alpha} - \delta g^{\alpha} (W^{\alpha}- W^0) +
\delta W^{\alpha},
 \end{equation}
 \begin{equation} \label{e15} 
 V^\beta = \overline{R} - R^{\alpha} - a^{\beta} + \delta f^\beta
(W^{\alpha} - W^0) + \delta W^\beta,
 \end{equation}
 and
\begin{equation} \label{e16} 
 V^0 = R^0 - a^0 + \delta f^0 (W^{\alpha}- W^0) - \delta g^0
(W^{\alpha} - W^0) + \delta W^0.
 \end{equation}

Our assumptions on $f$ and $g$ guarantee that an interior solution to
Alpha's problem exists.  The maximands (\ref{e14}) to (\ref{e16}) are
concave because $f$ and $-g$ are concave in $a$, so the first order
conditions  characterize the optimum.
  \begin{equation} \label{e17} 
 \frac{\partial V^{\alpha} }{\partial a^{\alpha}} = - 1 - \delta
\frac{\partial g}{\partial a^{\alpha}} (W^{\alpha} - W^0)= 0. 
 \end{equation}

\begin{equation} \label{e18} 
  \frac{ \partial V^\beta}{\partial a^{\beta}} = -1 + \delta
\frac{\partial f}{\partial a^{\beta}} (W^{\alpha} - W^0) = 0.
 \end{equation}

\begin{equation} \label{e19}
  \frac{\partial V^0}{\partial a^0} = -1 + \delta \frac{\partial
f}{\partial a^0} (W^{\alpha} - W^0) - \delta \frac{\partial g}{\partial
a^0} (W^{\alpha} - W^0) = 0.
 \end{equation}

 Note first that the assumptions of the model guarantee existence of
an equilibrium in pure strategies.

\noindent
 {\bf Lemma 1:} {In a one-period model or the last period of a
$T$-period model there exists a Nash equilibrium in pure strategies
in which each player exerts positive effort.}

\noindent
 {\bf Proof:}\\
 1. By a theorem from my book, pure-strategy equilibrium exists if
the strategy sets are compact, and payoff functions are continuous in
strategies of both players and quasi-concave in own-strategies. \\
 2. If we put an upper bound on effort, the strategy sets are
compact. Payoffs are continous, and are concave in own-effort.\\
 3. But in fact if the bound is high engouh, the solution is
interior.\\
 4. And if th solutin is interior, the bound can be removed without
makin any diffrence. 

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

\pagebreak


\noindent 
 {\bf Proposition 1:} {\it In a one-period model or the last period
of a $T$-period model, when one player is ahead his effort is
identical to the other player's effort and less than the effort when
the two players are even with each other, i.e.}
  \begin{equation}\label{e26}
 a^{\alpha} = a^{\beta}
  \end{equation}
{\it and}
  \begin{equation}\label{e27}
 a^{\alpha} < a^0.
  \end{equation}


\noindent
  {\bf Proof:} From first order conditions (\ref{e17}) and
(\ref{e18}), 
  \begin{equation} \label{e28} 
 - \frac{\partial g}{\partial a^{\alpha}} = \frac{\partial f}{\partial
a^{\beta}}.
 \end{equation}

 Suppose that Beta has the same effort when behind as when ahead, so
that $ b^{\alpha} = b^\beta$. Then assumption (\ref{e4})  tells us that
equality (\ref{e28}) is true only if $ a^{\alpha} = a^{\beta}$.  Since
Beta has first order conditions corresponding to Alpha's, it is
indeed rational for Beta to choose $ b^{\alpha} = b^\beta$ in a Nash
equilibrium, and equality (\ref{e26}) is proved.

To prove inequality (\ref{e27}), note from (\ref{e18}) and (\ref{e19}) that
  \begin{equation} \label{e29}
  \frac{\partial f}{\partial a^0} - \frac{\partial g}{\partial
a^0} = \frac{\partial f}{\partial a^{\beta}}.
 \end{equation}
 Given that $\frac{\partial g}{\partial a}<0$, equation (\ref{e29})
implies that 
  \begin{equation} \label{e30}
  \frac{\partial f}{\partial a^0} < \frac{\partial f}{\partial
a^{\beta}}.
 \end{equation}
 Given assumption (\ref{e3}) and the fact that $a^0=b^0$ by symmetry,
it follows from concavity of $f$ in $a$ that $a^0 > a^{\beta} =
 a^{\alpha}$. \\
  Q.E.D.

\bigskip

  Proposition 1 is interesting even aside from its relevance to
$T$-period games. Except for the unproductive nature of the
achievement in this model, the one-period game is like a 
one-stage patent race under uncertainty, with the complication that
the players might not start out even.  When one player gains what the
other loses, so they exert the same effort, and if the stakes are
higher they both exert a greater effort.

Equality (\ref{e26}) says that if one player is ahead, his effort is
the same as if he were behind, so by symmetry of the equilibrium
strategies, the follower exerts the same effort as the leader.  The
leader and the follower have exactly the same incentives. They are
fighting over the difference between being ahead and being even, the
leader defending a piece of property and the follower attacking it.
This is a special property of the one-period game, because in one
period the follower has no chance to become the leader, the leader
has no chance to become the follower, and neither need look ahead to
the cost of future effort.

 Inequality (\ref{e27}) says that both players exert more effort when
they are even than otherwise. If the two players are even, their
effort helps each one both defensively and offensively. Moreover, the
difference between worst-case failure and best-case success is at its
maximum for each player at a value of $V^{\alpha} - V^\beta$; more is
at stake.  If the players are not even, the leader's effort is purely
defensive and the follower's is purely offensive, so only $V^{\alpha} -
V^0$ is at stake.


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

\newpage
\noindent
 {\bf 4. THE GAME WITH TWO OR MORE PERIODS.}

   The two-period game is the simplest multi-period game, so let us
analyze it in some detail.  We must now be careful about attaching
time subscripts to variables.  Denote the last period, whose
characteristics were described in the preceding section, by the
subscript $T$.  Before starting to find the optimal strategies in the
first period of the two-period game, it is useful to know the
difference between the possible values in the second period.  Since
this difference will recur through the rest of the analysis, let us
define two new variables:
  \begin{equation} \label{e30a} 
 \begin{array}{l}
 D_t^{\alpha} \equiv  V_t^{\alpha} - V_t^0,\\ 
 D_t^0 \equiv  V_t^0 - V_t^\beta.\\ 
 \end{array}
 \end{equation}
 We can now succintly state Lemma 2, which will not only be useful
for the proof of Theorem 2, but has some independent interest, since
it states that in the last period the difference between being in the
lead and being even is greater than the difference between being even
and being behind.

  \noindent
 {\bf Lemma 2:} 
  \begin{equation} \label{e37} 
  D_T^{\alpha} > D_T^0.
   \end{equation}

\noindent
 {\bf Proof:}\\ 
 Using equations (\ref{e14}) and (\ref{e16}), one obtains
  \begin{equation} \label{e30aa} 
\begin{array}{lll}
 V_T^{\alpha}  - V_T^0   &=&  
  (R^{\alpha} - R^0)
   - (a_T^{\alpha} - a_T^0)
  - \delta g_T^{\alpha} (W^{\alpha} - W^0)
  - [ \delta f_T^0 (W^{\alpha} - W^0) \\
 & & - \delta g_T^0 (W^0 - W^\beta) ]
  +  \delta  W^{\alpha} -  \delta  W^0.\\
 \end{array}
 \end{equation}
 Since $W^{\alpha} -W^0 = W^0 -W^\beta$ by the symmetry assumption,
and $f_T^0 =g_T^0$ by Proposition 1, equation (\ref{e30aa}) becomes
  \begin{equation} \label{e31} 
 D_T^{\alpha} = (R^{\alpha} - R^0) - (a_T^{\alpha} - a_T^0) +
\delta (1- g_T^{\alpha}) (W^{\alpha} - W^0).
 \end{equation}
 Using equations (\ref{e14}) and (\ref{e15}), one obtains
  \begin{equation} \label{e31a} 
 \begin{array}{lll}
  V_T^0  - V_T^\beta   &=&  \\
  &=& 
  (R^0 - \overline{R} + R^{\alpha}) - (a_T^0 - a_T^\beta) + 
  \left[ \delta f_T^0 (W^{\alpha} - W^0)
- \delta  g_T^0 (W^0 - W^\beta) \right]\\ 
  & & - \delta f_T^\beta (W^0 - W^\beta)
  + \delta W^0 - \delta W^\beta.\\
  \end{array}
 \end{equation}
 Since by Proposition 1 $ (a_T^\beta= a_T^{\alpha}$ and $f_T^\beta
=g_T^{\alpha}$, equation (\ref{e31a}) becomes
  \begin{equation} \label{e32} 
  D_T^0 = (R^{\alpha} - R^0) + (a_T^{\alpha} - a_T^0) +\delta(1 -
g_T^\beta) (W^{\alpha} - W^0).
 \end{equation}
  Since $g_T^{\alpha} = f_T^\beta$ by Proposition 1, and $R^{\alpha} -
R^0 = R^0- R^\beta$, equations (\ref{e31}) and (\ref{e32}) can be
rewritten as
  \begin{equation} \label{e35}
 D_T^{\alpha} = \{(R^{\alpha} - R^0) + \delta (1-g_T^{\alpha}) (W^{\alpha} -
W^0)\} +\{ a_T^0 - a_T^{\alpha}\}
 \end{equation}
 and
  \begin{equation} \label{e36}
 D_T^0 = \{(R^{\alpha} - R^0) + \delta (1-g_T^{\alpha}) (W^{\alpha} -
W^0)\} - \{ a_T^0 - a_T^{\alpha}\}.
 \end{equation}
 Since Alpha's value is greater if the lead is greater, both
expressions (\ref{e35}) and (\ref{e36}) are positive.  By Proposition
1, $\{a_T^0 - a_T^{\alpha}\} > 0$.  Comparing equation (\ref{e35}) with
(\ref{e36}) therefore tells us that $ D_T^{\alpha} > D_T^0$.\\
 Q.E.D. 

\bigskip

   Now the problem of finding the equilibrium strategies in the first
period can be tackled. The values of Alpha in the three possible
states are
  \begin{equation} \label{e38} 
 V_{T-1}^{\alpha} = R^{\alpha} - a_{T-1}^{\alpha} - \delta g_{T-1}^{\alpha}
D_T^{\alpha} + \delta V_T^{\alpha},
 \end{equation}
  \begin{equation} \label{e39} 
  V_{T-1}^\beta = R^\beta - a_{T-1}^\beta + \delta f_{T-1}^\beta
D_T^\beta + \delta V_T^\beta,
 \end{equation}
 and
  \begin{equation} \label{ee40} 
 V_{T-1}^0 = R^0 - a_{T-1}^0 + \delta f_{T-1}^0 D_T^{\alpha} -\delta
g_{T-1}^0 D_T^0 + \delta V_T^0.
 \end{equation}
  The first order conditions are (recalling that that $D_T^{\alpha}$
and $D_T^\beta$ are independent of the effort levels in the first
period)
  \begin{equation} \label{ee44} 
  \frac{\partial V_{T-1}^{\alpha}}{\partial a_{T-1}^{\alpha}} = - 1 -
\delta \frac{\partial g}{\partial a_{ T-1}^{\alpha} } D_T^{\alpha} = 0, 
 \end{equation}
 \begin{equation} \label{ee45} 
  \frac{ \partial V_{T-1}^\beta}{\partial a_{T-1}^\beta} = -1 + \delta
\frac{\partial f} {\partial a_{ T-1}^\beta } D_T^0 =0,
 \end{equation}
 and
 \begin{equation} \label{ee46}
 \begin{array}{ll}
  \frac{\partial V_{T-1}^0}{\partial a_{T-1}^0} & = -1 + \delta
\frac{\partial f}{\partial a_{ T-1}^0} D_T^{\alpha} - \delta \frac{\partial
g}{\partial a_{ T-1}^0} D_T^0 = 0.\\
 \end{array}
 \end{equation}
 These first order conditions can be used to prove Proposition 2.


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

\newpage

\bigskip
\noindent 
 {\bf Proposition 2:} {\it In the next-to-last period each player
exerts greater effort when even with the other player than if one of
them is ahead. If one of them is ahead, the leader exerts greater
effort than the follower. Therefore,}
  \begin{equation}\label{ee50}
 a_{T-1}^{\alpha} < a_{T-1}^0
  \end{equation}
 and
  \begin{equation}\label{ee51} 
 a_{T-1}^\beta < a_{T-1}^{\alpha}.
  \end{equation}

\noindent
 {\bf Proof:} To prove inequality (\ref{ee50}), note that first order
conditions (\ref{ee44}) and (\ref{ee46}) imply that
  \begin{equation} \label{ee53}
 - \frac{\partial g}{\partial a_{T-1}^{\alpha}} D_T^{\alpha} =
\frac{\partial f}{\partial a_{T-1}^0} D_T^{\alpha} - \frac{\partial
g}{\partial a_{T-1}^0 } D_T^0.
 \end{equation}
 Since in equilibrium $ a_{T-1}^0= b_{T-1}^0$ by symmetry, assumption
(\ref{e4}) implies that $\frac{\partial f}{\partial a_{T-1}^0} =
-\frac{\partial g}{\partial a_{T-1}^0}$. Hence one may rewrite
(\ref{ee53}) as 
  \begin{equation} \label{ee54}
 - \frac{\partial g}{\partial a_{T-1}^{\alpha}} D_T^{\alpha} = 
 - \frac{\partial g}{\partial a_{T-1}^0 } (D_T^{\alpha} + D_T^0),
 \end{equation}
 in which case
  \begin{equation} \label{ee55}
 - \frac{\partial g}{\partial a_{T-1}^{\alpha}} >
 - \frac{\partial g}{\partial a_{T-1}^0 },
 \end{equation}
 which implies, given assumption (\ref{e3}) that $\frac{
\partial^2f}{\partial a \partial b} \geq 0$, that $a_{T-1}^0 >
a_{T-1}^{\alpha}$.

  To prove inequality (\ref{ee51}), note that since $D_T^{\alpha} >
D_T^0$, first order conditions (\ref{ee44}) and (\ref{ee45}) imply
that
 \begin{equation} \label{ee52} 
 -\frac{\partial g}{\partial a_{T-1}^{\alpha}} < \frac{\partial
f}{\partial a_{T-1}^\beta}.
 \end{equation}

 By assumption (\ref{e5}) and the assumption that the equilibrium is
symmetric, $ -\frac{\partial g}{\partial a_{T-1}^{\alpha}} =
-\frac{\partial f}{\partial b_{T-1}^\beta}$, so equation (\ref{ee52})
implies
  \begin{equation} \label{ee52a}
  -\frac{\partial f}{\partial b_{T-1}^\beta} < \frac{\partial
f}{\partial a_{T-1}^\beta}.
 \end{equation}

By assumption (\ref{e4}), if $ a_{T-1}^\beta = b_{T-1}^\beta$ then
expression (\ref{ee52a}) would be an equality.  Since it is an
inequality, the concavity of $f$ in $a$ combined with assumption
(\ref{e5}) and symmetry tells us that $ a_{T-1}^\beta <
b_{T-1}^\beta$. In equilibrium, $ b_{T-1}^\beta= a_{T-1}^{\alpha}$, so
(\ref{ee51}) is proven.\\
 Q.E.D.

\bigskip

   Propositions 1 and 2 show that players behave differently in the
first and second periods of a game.  The proof method of Proposition
2 can be applied recursively to show that it is the last period that is
special, not the first; the inequalities of Proposition 2 apply to
all but the last period. This is stated in Proposition 3, whose proof
is the Appendix. 

\bigskip
  \noindent 
 {\bf Proposition 3:} {\it In all but the last period of a $T$-period
game, each player exerts greater effort when even with the other
player than if one of them is ahead. If one of them is ahead, the
leader exerts greater effort than the follower. For every $t<T$,}
  \begin{equation}\label{e50a}
 a_t^{\alpha} < a_t^0.
  \end{equation}
  {\it and}
  \begin{equation}\label{e50}
a_t^\beta < a_t^{\alpha}
  \end{equation}

\bigskip
\noindent
 {\bf Proof:} See the Appendix.

 Note the implications here for the amount of time spent in each
state. As T gets large, we can predict that more time will be spent
in states with one ahead or behind.


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


\newpage 

\noindent
{\bf 5. RELAXING ASSUMPTIONS.}

  It may not be easy to tell which assumptions are important in
generating Propositions 1 to 3. In this section I will discuss the
properties of the model and consider relaxing certain of the
assumptions. 


\noindent
 {\bf Intuition behind the propositions.}

  Section 3 explained why in the last period the leader's effort
level is the same as the follower's: what one gains, the other loses.
Propositions 2 and 3, however, say that in every period but the last,
the leader exerts more effort.  The reason is that expected future
effort affects the choice of current effort.  Consider a two-period
game.  If effort levels were always zero in the second period, then
in the first period the leader and the follower would have identical
incentives and exert the same effort.  But effort levels in the
second period are higher if the players are even than if somebody is
ahead.  When the follower succeeds in becoming even, he gains in
benefit, but his second period effort increases too.  When the leader
successfully stays ahead, he both preserves his benefit level and
avoids increasing his second period effort. The leader's incentive to
exert effort is therefore greater than the follower's.

 In every period, including the last, the players exert the most
effort when they are even. In the last period, this is because when
the players are even, the possible changes in position are greatest.
When a player can either advance or regress, research is useful for
both offense and defense. The same intuition applies in the earlier
periods, but it is reinforced by the effect described in the last
paragraph.  Moving away from being even lowers future effort, while
moving away from being the leader or the follower increases future
effort.


  The explicit assumptions were discussed as they arose in describing
the model, but more can be said about some of the more implicit
assumptions.

\bigskip
\noindent
  {\bf Three States, not $N$. } The assumption that a player can only
be ahead, behind, or even is a simplifying assumption.  Although the
complexity and notational difficulty increases rapidly, the intuition
behind the propositions continues to hold with more states.  Consider
a leader whose effort can increase his share of the benefit instead
of just defending it.  His share cannot increase indefinitely, since
it is bounded by one hundred percent, so either (a) his effort
eventually becomes purely defensive or (b) the offensive component of
his incentive becomes smaller as the bound is approached.  In either
case, there is less at stake for the leader as he pulls further
ahead. Since the benefit side of the game is zero-sum, this implies
that there is less at stake for the follower. If this verbal
reasoning is correct, effort will be greater when the lead is closer
to zero, and this implies that the leader has more incentives than
the follower.  \footnote{xxx Cite Harris paper here.}


\bigskip
\noindent
  {\bf One-Level Transitions, not Two-Level.} An implicit assumption
is that change in the lead is gradual: there is not a jump from being
ahead to being behind without the transition of being even. One might
wonder whether this assumption drives the result that effort is
highest when the firms are even.  Indeed it does, but the assumption
can be relaxed considerably without losing the result. In the
extreme, suppose that the function $f$ represents not the probability
of a change of one level in the lead, but rather the probability of
Alpha gaining the lead. If Alpha starts out behind, $f$ is the
probability of gaining two levels, while if Alpha starts out even,
$f$ is the probability of gaining one level.$^4$ The first order
condition for Alpha when he starts with a lead changes only in the
substitution of $W^\beta$ for $W^0$.  Equation (\ref{e17}) is
transformed to
  \begin{equation} \label{e200} 
 \frac{\partial V^{\alpha} }{\partial a^{\alpha}} = - 1 - \delta
\frac{\partial g}{\partial a^{\alpha}} (W^{\alpha} - W^\beta)= 0. 
 \end{equation}
  But this substitution is enough that in combination with the
unchanged first order condition (\ref{e19}) an equivalent to
(\ref{e29}) can be generated: 
 \begin{equation} \label{e219}
 \frac{\partial f}{\partial a^0} (W^{\alpha} - W^0) -
\frac{\partial g}{\partial a^0} (W^{\alpha} - W^0) = \frac{\partial
g}{\partial a^{\alpha}} (W^{\alpha} - W^\beta),  
 \end{equation}
 which implies, because $ \frac{\partial f}{\partial a^0}=
-\frac{\partial g}{\partial a^0}$ and $(W^{\alpha} - W^\beta) = 2
(W^{\alpha} - W^0)$, that
 \begin{equation} \label{e221}
 -2\frac{\partial g}{\partial a^0} (W^{\alpha} - W^0) = -\frac{\partial
g}{\partial a^{\alpha}} 2(W^{\alpha} - W^0), 
 \end{equation}
 which implies that $a^{\alpha} = a^0$. In this extreme case, the
results in Propositions 1 to 3 disappear. In a given period the
players exert the same effort regardless of the state, although that
effort may vary as time passes. But an intermediate model that broke
up $f$ into a probability $\theta f$ that Alpha's lead increases by
one level and a probability $(1-\theta)f$ that it increases by two
would restore the conclusion that $a^0 > a^{\alpha}$, with the size of
$a^0 - a^{\alpha}$ depending on $\theta$. 


 \bigskip 
 \noindent 
  {\bf A Transition Functions with Three Values, not Two.} Another
modification is to define $f$ to be the probability of an increase in
the lead and $(1-f)$ to be the probability of a decrease.  If a
period started with the players even, either Alpha or Beta would be
ahead at the end of the period: stasis is not allowed. All the
assumptions of the original model concerning $f$ are still required,
and the results are exactly the same: Propositions 1, 2, and 3 remain
true.

 Being even is a transitory state in the two-step transition model.
The model might start with both players even, but it never returns to
that state.

\bigskip 
 \noindent 
 {\bf $T$ periods, not an Infinite Number.} The assumption that the
number of periods is finite is important for modelling reasons.
Proposition 3 takes us most of the way to an infinite period model,
because $T$ can be great enough that the last period exerts a trivial
effect. The Folk Theorem of repeated games, however, tells us that in
a wide variety of games with infinite repetitions and sufficiently
little discounting, a large number of equilibria exist (see Fudenberg
\& Maskin [1986]).

  The rivalry model in this paper is not a stationary repeated game,
because the game can be in any of three states.  This nonstationarity
means that the Folk Theorem does not apply directly. But the
reasoning behind the Folk Theorem--- that with sufficiently little
discounting, arbitrary equilibrium behavior can be enforced by the
threat of future punishments--- can be made to apply.  The punishment
would take the form of very high effort by the punisher for a large
number of periods, and if the punisher deviated by not carrying out
the punishment, which is costly for him, the strategy would specify
that he himself be punished by the other player.  As a result of this
argument, the infinite period game is not so interesting as the
finite period game.  The outcomes characterized in the main part of
the paper can be supported by equilibria in both games, but the
infinite period game is underdetermined and has other equilibria as
well .

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

\newpage
\noindent
 {\bf 6. THE RIVALRY LITERATURE.}

   Outside of the literature on technical change, papers which ask
questions about rivalry include J. Hirshleifer (1987), Aron \& Lazear
(1987), and Dixit (1987). Hirshleifer analyzes the choice between
fighting and producing, a choice that is not available in the model
here.  Aron and Lazear ask whether firms that are leaders in market
share in the current market are less likely to switch to a new
product line. This is an example of rivalry, but the emphasis is not
on the intensity of a single rivalrous action, but on the variety of
rivalrous actions available.

  Dixit (1987) examines pure rivalry in a one-period contest in which
both players start at the same level, but the progress function might
give one player an advantage. If the progress function is symmetric,
Dixit shares the conclusion of this paper that both players exert the
same effort. The present paper has ruled out asymmetric progress
functions, but it it interesting to note that Dixit finds that the
player with the advantage (that is, who is more likely to win if
effort levels are the same) exerts greater effort. This is proved for
particular functional forms, and the reason behind the result is
completely different from the reason in the multi-period model of
rivalry. \footnote{xxx what is that reasons?}


\bigskip
\noindent
 {\bf The Innovation Literature.}


Are Lippman-Mccardle and Harris-Vickers(1985) relevant? 
 Is Beath(1987) relevant?

XW

 Rivalry has been most studied in the context of innovation. Most of
the literature analyzes single-innovation models, in which research
ends after the innovation is found. This can describe rivalry over
time if the research process has many stages (see Fudenberg, Gilbert,
Stiglitz \& Tirole [1983], Harris \& Vickers [1985], and Lippman \&
McCardle [1987]), but a patent is special in the sense that one
player gets it and the other player does not; how close the loser
comes does not matter.

 The Avis Question came up first in the patent-race models of Gilbert
\& Newbery (1982) and Reinganum (1985). Gilbert and Newbery construct
a model of sleeping patents in which payoffs may be non-zero-sum even
on the benefit side.  One firm has a monopoly on a technology and
another threatens to discover a substitute. The monopolist does
research to try to patenting the substitute first, whether he intends
to use it or not, and he spends more than his rival if a monopoly
with access to both technologies could earn higher profits than a
duopoly.  Reinganum (1985) models a sequence of patentable
innovations and reaches a conclusion opposite to both Gilbert \&
Newbery and the present paper: the incumbent spends less on research
than the rivals.  This is because the time between innovations is
endogenous.  The expected cost of discovering the next innovation in
the sequence is equal for each firm in this model, so once a
particular patent race ends, no firm enjoys any advantage in
discovering the next patent.  The incumbent spends less on research
because the net revenue flow from the new technology is the same
whichever firm discovers it, but because the incumbent loses the
revenue flow from the old technology, his net benefit from innovation
is less.  The endogeneity of the interval between discoveries is
crucial: the incumbent has less or no incentive to hurry the
discovery. The biggest difference from the present paper, however, is
that in the Reinganum model the incumbent has no advantage over the
other firms in the struggle for the future incumbency; he is just an
incumbent, not a leader.  Which firm makes the next innovation is
thus irrelevant to the future amount of research.

The model of Vickers (1986) can be considered an application of this
idea when time periods are exogenous and the innovation is not
``drastic''. There is a sequence of falling cost levels that firms
move throguh.  Two firms. T periods. There is a new lower cost level
for each period, and the players bid for who gets to have that cost
level. No discounting, but that does not matter. Certainty.  You can
jump ahead with no problem. The outcome is action-recton if industry
prfits are higher when the follower's costs fall. If the high-cost
firm gets zero profits, we gte increasing domiancne. 


  A number of recent models have looked at the Action-Reaction vs.
Increasing Dominance question in more detail. These models see
rivalry as progress along a sequence of achievement levels, which in
an innovation model would consist of different products or
technologies. The paper closest to the present paper are Grossman \&
Shapiro (1987), Harris \& Vickers (1987), and Harris (1988). 

Grossman \& Shapiro (1987) look at a multi-stage patent discovery
process, in which there is only one innovation to be found, which
will not be found without research. The research technology is an
arrival process, and the game ends once one player reaches the final
stage of the patent search. Their conclusions on which firm exerts
more effort are consistent with the predictions of the present paper,
but the conclusions are driven by different forces. They find that
rivals spend more on research when they are even for two reasons: the
potential change in values is then greatest, and the reaction curves
are upward sloping. The first of these reasons drives the same result
in the present paper.  Grossman and Shapiro also find that the leader
spends more than the follower, but their reason is not the
expectation of future costs, but that the leader is closer to the
finish line of the patent race.



 Other papers have dealt with other aspects of sequential innovation.
Beath, Katsoulacos, \& Ulph (1987) focus on how the speed of the
innovation technology and the form of competition in the product
market affect research spending to find a patent. These also include
Beath, Katsoulacos, \& Ulph (1989) and Aoki 88.  Aoki (1988) uses the
framework of a stochastic game with an infinite number of periods.
The follower can earn profits only if his technology is close enough
to the leader's, and there are three possible profit levels: zero,
duopoly, and monopoly.


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

\newpage
\noindent
{\bf 7. APPLICATIONS.}

 The model is general enough to apply to many situations of rivalry.
I will list some such situations below, with suggestions as to what
would represent effort, the lead, the current benefits, and the
balloon benefit.  The model has made ruthless use of the {\it ceteris
paribus} assumption, so one would not expect it to fully explain any
single situation, but it does point out effects of rivalry that are
common to many situations.  These effects are superimposed on other
tendencies that may reverse the predictions of the model. The model
itself has different predictions depending on whether the situation
is best modelled as a one period game or as a multiperiod game. In
the discussion below I will treat the situations as multiperiod
games, which assumes that the players have a chance to revise their
effort levels after observing the effect of their earlier efforts.

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

\noindent 
 {\bf (1) Innovation.} In an innovative industry, a firm's market
share depends on the quality of its product relative to its
competitor's product. To maintain relative quality, a firm must spend
on research, but success in research is uncertain, and a follower
firm does have a chance to catch up to the leader. In some cases the
benefit will be the profits from current production, but in other
cases there may be an additional balloon benefit if at some point the
leading firm becomes able to maintain its sales even if it slackens
it research; e.g.  if there are network externalities. The rivalry
model predicts that research spending will be highest when two firms
split the market evenly, and that if one firm is ahead, that firm
will spend the most on research.


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

\noindent 
 {\bf (2) Wars.} In a war, two nations exert effort to capture a
resource whose value does not increase if they fight more intensely,
and may even decrease.  The lead is the probability of winning a
complete victory, and effort is the amount of resources spent on the
war.  The benefits of being ahead are likely to be small while the
war continues, but at the end the country which is ahead can acquire
territory, influence over the government of the vanquished, or
influence over third countries. A difference from the rivalry model
is that the end date of a war is endogenous.  The model predicts that
a nation that is winning will fight harder than a nation that is
losing, because the losing nation's successes in individual battles
prolong the suffering of war. The effort exerted will be most intense
when neither side is clearly winning.

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

\bigskip
 \noindent
 {\bf (3) Arms Races.} In an arms race, two countries strive to
increase their influence by military superiority that they may never
directly use. The difference from a war is that the cost is purely in
dollars rather than lives lost, and that an arms race is less likely
to end. One form of arms race--- the race to have the best arms
technology by spending on research--- is practically identical to the
rivalry model.  What is more generally thought of as an arms race---
spending on the stock of weapons--- fits the rivalry model if it is
uncertain how this spending translates into gains in power.

    The lead in this second kind of arms race is the likelihood of
winning a war, should one break out, and the effort is the amount
spent on arms.  The current payoffs are the amounts of influence
derived from the threat to start and win a war, and the final period
can be far enough away that the balloon payoff is unimportant.  The
rivalry model's prediction is that the nation which is winning the
arms race will spend more than the loser, because the loser's
successes increase future spending on arms. Both nations will spend
more when their arms levels are close than when the gap is large.
Note, however, that implicit in the rivalry model is the stationarity
of the functions translating effort into gains. If accumulating more
than a certain number of warheads, for example, fails to increase a
nation's power, the model's assumptions are invalid and one would not
expect the leader to accumulate more than that number. 

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

\bigskip
 \noindent
 {\bf (4) Advertising.} The rivalry model fits advertising in the
same way as it fits innovation, since what is important about a
product is its profitability, whether that is derived from
technological excellence or consumer perceptions. Although the
assumption of a market of constant size is not crucial to the model,
it does seem more applicable to an advertising model than an
innovation model. The lead is a measure of buyer preference, and
effort is spending on advertising.  The model predicts that the firm
with the biggest market share will spend most heavily on advertising,
and that both firms will spend more when their market shares are
equal.

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

\bigskip
 \noindent
 {\bf (5) Pecking Orders.} Humans and lower animals exert effort to
maintain and improve their positions in pecking orders. A pecking
order can be useful to a social group, but the benefit to an
individual who moves up is largely a loss to the individual who moves
down.  The lead is position in the pecking order, and effort could be
anything from physical combat to achievements that benefit and
impress the rest of the group. The rivalry model predicts that close
rivals would exert more effort, and that the dominant individual
would be the more willing to prolong a power struggle. 

 The rivalry model also has implications for why a species might
benefit from (a) dominance hierarchies and (b) the emotion of
demoralization.  Dominance hierarchies are useful because they
diminish the wasteful expenditure of effort that the rivalry model
implies would occur in a group of equals. Demoralization is helpful
because the loser in a struggle is benefited by diminishing his
effort, whether he diminishes it for rational or emotional reasons.
If the rivalry model is correct, one would therefore expect
evolutionary pressures to lead to hierarchies and demoralization.

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

\bigskip
 \noindent
 {\bf (6) Internal Politics.} Suppose that each member of an
organization lends some amount of support to each of the two factions
into which it is divided.  Applying the rivalry model, the lead can
be viewed as the amount by which a given member (most importantly,
the organization's leader) supports one faction, and effort is the
amount of time and resources spent trying to persuade him to change
his support. The model predicts that the faction currently
controlling a member is more willing to exert effort to keep him, but
that even more effort would be spent if he were undecided.


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

\bigskip
 \noindent
 {\bf (7) Elections.} A reasonable simplification is that an election
campaign yields no current benefits to the two candidates, but on
election day whoever is ahead is paid a balloon benefit. Ties are
unlikely, so the model with the two-valued transition function fits
the situation best.  The model predicts that campaign spending will
be highest when the candidates are running even in the polls, and
that if one candidate is ahead, he will spend more than his lagging
rival.

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

\bigskip
 \noindent
 {\bf (8) Artificial Tournaments.}
 A tournament is an incentive mechanism that distributed rewards
according to relative performance instead of absolute performance,
for performing best rather than for performing well.  From the work
of Lazear \& Rosen (1981), Nalebuff \& Stiglitz (1983), and others,
it is well-known that principals can find tournaments helpful in
overcoming agency problems. Examples include awarding a profitable
contract to the aircraft company that comes up with the best jet
fighter design by a certain date, giving tenure to the assistant
professor who has the best research output in the department, and
presenting a company award to the industrial plant with the best
output or safety record during a particular month.

  It has already been established that a tournament will not work
well if the contestants differ too much in ability. The rivalry model
adds a dynamic counterpart to that idea: the tournament will induce
the most effort if the contestants happen to stay even with each
other over the time period in which the tournament takes place.
Whether the designer of a tournament can make use of this knowledge
is a question too involved to try to answer here.

\bigskip
 \noindent
 {\bf (9) Rentseeking.}
   Rentseeking is halfway between the economic and military examples:
economic agents expend effort to induce the government, with its
monopoly of force, to change the economic system in their favor.
Coalitions are often important in the political process of
rentseeking (see, for example, Gilligan et al. [forthcoming]), but
sometimes rentseeking is simple rivalry between firms bidding for
government help against consumers, or buyers and sellers bidding for
government help against each other.

  The rivalry model implies that if one group is succeeding in
capturing a government body, then it will exert more effort than its
competitor. Suppose, for example, that the opinion of the chairman of
a congressional transportation committee is affected both by campaign
contributions and by random factors such as the merits of an issue.
According to the model, both the railroad and the trucking trade
associations would always contribute to him, but if he has become
pro-railroad, the railroads would contribute more. If the congressman
were undecided, he would attract still greater contributions (though
in this model his stance is endogenous, so he cannot choose to be
undecided).  


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


\bigskip
 \noindent
 {\bf Concluding Remarks.}

  The model has shown that under the assumptions of pure rivalry, in
most periods the leader exerts more effort than the follower, but
both exert less effort than if they were even with each other.  An
exception is the last period, or the only period of a one-period
model, in which the leader and the follower exert the same effort.

 Each part of the result has an intuitive explanation.  The leader's
success reduces the cost of future competition, while the follower's
success increases it, so the leader has more incentive to exert
effort. When the rivals are even, the effect of a change is greater,
so both players have more incentive. And in the last period, what one
player gains, the other loses, so the incentives are evenly balanced.


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

\newpage

\begin{center}
{\bf Appendix.}\\
\end{center}

 The appendix contains the proof of Proposition 3. It is convenient
to first prove a lemma regarding different states' valuations in the
two-period game.

  This next lemma has a flawed proof, and it is crucial. The problem
is that in period T-1 the big thing is to avoid high costs in period
T; which is to avoid being in state 0 at T. But maybe starting in
state 0 at T-1 is the best ...

The key thing is to show that the probability of remaining in an
extreme state is greater than the probability of going to an extreme
state from the 0 state. And that, I think , is impossible to rpovie.
Alternately, and easier, get rid of the balloon payment, and then we
can prove that spending is no greataer in the last period.  


\bigskip
\noindent 
 {\bf Lemma 3:} {\it If $ D_{t}^{\alpha} > D_{t}^0$, then
$D_{t-1}^{\alpha} > D_{t-1}^0$.}

\noindent 
 {\bf Proof:} Generalizing equations (\ref{e38}) to (\ref{ee40}) from
a two-period model to an $n$-period model, we obtain
  \begin{equation} \label{e51} 
 V_t^{\alpha} = R^{\alpha} - a_t^{\alpha} - \delta g_t^{\alpha}
D_{t+1}^{\alpha} + \delta V_{t+1}^{\alpha},
 \end{equation}
 \begin{equation} \label{e52} 
  V_t^\beta = R^\beta - a_t^\beta + \delta f_t^\beta D_{t+1}^0 +
\delta V_{t+1}^\beta,
 \end{equation}
 and
\begin{equation} \label{e53} 
 V_t^0 = R^0 - a_t^0 + \delta f_t^0 D_{t+1}^{\alpha}  - \delta
 g_t^0 (D_{t+1}^0 + \delta V_{t+1}^0.
 \end{equation}
 Differentiating, one obtains the first order conditions
 \begin{equation} \label{e54} 
  \frac{\partial V_t^{\alpha}}{\partial a_t^{\alpha}} = - 1 - \delta
\frac{\partial g}{\partial a_t^{\alpha}} D_{t+1}^{\alpha} = 0, 
 \end{equation} 
 \begin{equation} \label{e55} 
  \frac{ \partial V_t^\beta}{\partial a_t^\beta} = -1 + \delta
\frac{\partial f}{\partial a_t^\beta} D_{t+1}^0 =0,
 \end{equation}
 and
\begin{equation} \label{e56}
 \begin{array}{ll}
  \frac{\partial V_t^0}{\partial a_t^0} & = -1 + \delta
\frac{\partial f}{\partial a_t^0} D_{t+1}^{\alpha} - \delta
\frac{\partial g}{\partial a_t^0} D_{t+1}^0 =0.\\
 \end{array}
 \end{equation}

 Using equations (\ref{e51}) to (\ref{e53}), one obtains
  \begin{equation} \label{e44} 
 \begin{array}{ll}
  D_{t-1}^{\alpha} & = R^{\alpha} - R^0 - (a_{t-1}^{\alpha} - a_{t-1}^0)
- \delta g_{t-1}^{\alpha} D_t^{\alpha} + \delta V_t^{\alpha} - \delta
f_{t-1}^0 D_t^{\alpha} + \delta g_{t-1}^0 D_t^0 - \delta V_t^0,\\ 
 & \\
 & = R^{\alpha} - R^0 + (a_{t-1}^0 - a_{t-1}^{\alpha}) + \delta (1 -
g_{t-1}^{\alpha}) D_t^{\alpha} - \delta f_{t-1}^0 D_t^{\alpha} + \delta
g_{t-1}^0 D_t^0\\
 \end{array}
 \end{equation}
  and
  \begin{equation} \label{e45} 
\begin{array}{ll}
  D_{t-1}^0 = & R^0 - R^\beta - (a_{t-1}^0 - a_{t-1}^\beta) + \delta
f_{t-1}^0 D_t^{\alpha} - \delta g_{t-1}^0 D_t^0 + \delta V_t^0 - \delta
f_{t-1}^\beta D_t^0 - \delta V_t^\beta,\\ 
 & \\
 & = R^0 - R^\beta - (a_{t-1}^0 - a_{t-1}^\beta) + \delta (1 -
f_{t-1}^\beta) D_t^0 - \delta g_{t-1}^0 D_t^0 + \delta f_{t-1}^0
D_t^{\alpha}.
 \end{array}
 \end{equation}

 Using  equations (\ref{e44}) and (\ref{e45}):
  \begin{equation} \label{e48} 
 \begin{array}{ll}
  D_{t-1}^{\alpha} - D_{t-1}^0 = & 2a_{t-1}^0 - a_{t-1}^{\alpha}-
a_{t-1}^\beta\\
  & + \delta \{ (1 - g_{t-1}^{\alpha}) D_t^{\alpha} - f_{t-1}^0
D_t^{\alpha} + g_{t-1}^0 D_t^0 - (1 - f_{t-1}^\beta) D_t^0 + g_{t-1}^0
D_t^0 - f_{t-1}^0 D_t^{\alpha} \}.
 \end{array}
 \end{equation}
 By symmetry of the equilibrium, $g_{t-1}^{\alpha}=f_{t-1}^\beta$ and
$f_{t-1}^0=g_{t-1}^0$, so we can rewrite equation (\ref{e48}) as 
  \begin{equation} \label{e49} 
  D_{t-1}^{\alpha} - D_{t-1}^0 = \{2a_{t-1}^0 - a_{t-1}^{\alpha}-
a_{t-1}^\beta \} + \delta \{ (1 - g_{t-1}^{\alpha} - 2g_{t-1}^0)
(D_t^{\alpha}- D_t^0) \}.
 \end{equation}

  Because probabilities must add up to one, and because Proposition 
 1 tells us that $f_{t-1}^{\alpha} >g_{t-1}^{\alpha}$ and $g_{t-1}^0)=
f_{t-1}^0)$, we know that $g_{t-1}^{\alpha}$ and $g_{t-1}^0)$ are each
no greater than 0.5 in equilibrium.  Hence, $\delta (1 -
g_{t-1}^{\alpha} - 2g_{t-1}^0) \in [-.5, 1]$. If that expression is
positive, then expression (\ref{e49}) is positive and the proposition
is proved. If it is negative, then the proposition is proved if it
can be shown that
 \begin{equation} \label{e49aa}
 (D_t^{\alpha}- D_t^0) \leq \{2a_{t-1}^0 - a_{t-1}^{\alpha}-
a_{t-1}^\beta \}. 
 \end{equation}

 We can prove Lemma 3 for $t=$, because for $t=T$ it turns out that
 \begin{equation} \label{e49a}
 D_T^{\alpha}- D_T^0 = 2a_T^0 - a_T^{\alpha} - a_T^\beta.  
 \end{equation}
  By substituting a value taken from first order condition
(\ref{e55}) for the 1 in first order condition (\ref{e56}), and then
dividing by $\delta$, we obtain
 \begin{equation} \label{e49b} 
 -  \frac{\partial f}{\partial a_t^\beta} D_{t+1}^0 
 +  \frac{\partial f}{\partial a_t^0} D_{t+1}^{\alpha} - 
\frac{\partial g}{\partial a_t^0} D_{t+1}^0 =0.
 \end{equation}
 Recognizing that by symmetry $-\frac{\partial g}{\partial a_t^0} =
\frac{\partial f}{\partial a_t^0}$, equation (\ref{e49b} becomes 
 \begin{equation} \label{e49c} 
   \frac{\partial f}{\partial a_t^\beta} D_{t+1}^0= 
 + \frac{\partial f}{\partial a_t^0} (D_{t+1}^{\alpha} + D_{t+1}^0).
 \end{equation}
 With a little rearranging, we obtain
 \begin{equation} \label{e49d} 
 \frac{  \frac{\partial f}{\partial a_t^\beta}}
 { \frac{\partial f}{\partial a_t^0}} = 
  1 + \frac{ D_{t+1}^{\alpha}}{D_{t+1}^0}.
 \end{equation}
 Consider the second part of the right-hand-side of equation
(\ref{e49d}). For $t=T$, that second part equals 1. For $t=T-1$, that
second part equals something greater than 1, by Lemma 2. Hence, the
left-hand-side of equation (\ref{e49d}) is greater for $t=T-1$ than
for $t=T$, and we have shown that $(a_{T-1}^0- a_{T-1}^\beta)
>(a_{T}^0- a_{T}^\beta)$. We could go through a parallel analysis to
show the analogous result for $a^{\beta}$. It follows from equation
(\ref{e49a}) that inequality (\ref{e49aa}) is true and the
proposition is true for $t=T$, and so $D_{T-1}^{\alpha} > D_{T-1}^0$.
This would be enough to prove Proposition 3 for $t=T-2$.

It remains to show (\ref{e49aa}) is true more generally.\\
  Q.E.D.


\bigskip
  \noindent 
 {\bf Proposition 3:} {\it In all but the last period of a $t$-period
game, each player exerts greater effort when even with the other
player than if one of them is ahead. If one of them is ahead, the
leader exerts greater effort than the follower. For every $t<T$,}
 $$
\begin{array}{ll}
 (\ref{e50a}) &  a_t^{\alpha} < a_t^0
 \end{array}
 $$
  {\it and}
 $$
 \begin{array}{ll}
 (\ref{e50}) & a_t^\beta < a_t^{\alpha}.
 \end{array}
 $$


\bigskip
\noindent
 {\bf Proof:} The proof works by recursion, and has four parts.

\bigskip
\noindent
{\bf Part II:  Inequality (\ref{e50a}).}\\
  To prove inequality (\ref{e50a}), note that first order conditions
(\ref{e54}) and (\ref{e56}) imply that
  \begin{equation} \label{e59}
 - \frac{\partial g}{\partial a_t^{\alpha}} D_{t+1}^{\alpha} =
 \frac{\partial f}{\partial a_t^0} D_{t+1}^{\alpha} - \frac{\partial
g}{\partial a_t^0} D_{t+1}^0. 
 \end{equation}

  Since in equilibrium $ a_t^0= b_t^0$ by symmetry, assumption
(\ref{e4}) implies that $\frac{\partial f}{\partial a_t^0} =
-\frac{\partial g}{\partial a_t^0}$. Hence 
(\ref{e59}) can be rewritten as 
  \begin{equation} \label{e60}
\begin{array}{ll}
 - \frac{\partial g}{\partial a_t^{\alpha}} D_{t+1}^{\alpha} & =
-D_{t+1}^{\alpha} + D_{t+1}^0 \frac{\partial g}{\partial a_t^0}\\ 
 & =- D_{t+1}^{\alpha} \frac{\partial g}{\partial a_t^0}. 
  \end{array}
 \end{equation}

which implies that
  \begin{equation} \label{e61}
 - \frac{\partial g}{\partial a_t^{\alpha}} > -\frac{\partial
g}{\partial a_t^0}, 
 \end{equation}
  which implies, given assumption (\ref{e3}), that $a_t^0 >
a_t^{\alpha}$, for any $t$.

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


\bigskip 
 \noindent 
 {\bf Part III:  Inequality (\ref{e50}) for t = T-2.}\\
 Proposition 2 tells us that Proposition 3 holds for $t=T-1$, since
in a perfect equilibrium the subgame consisting of the last two
periods of a T-period game has the same equilibrium as a two-period
game.  What must be shown is that Proposition 3 also holds for
earlier periods. Let us start with $t = T-2$.

 To prove inequality (\ref{e50}) for $t = T-2$, note that first order
conditions (\ref{e54}) and (\ref{e55}) imply that
  \begin{equation} \label{e57} 
 - \frac{\partial g}{\partial a_t^{\alpha}} D_{t+1}^{\alpha} =
\frac{\partial f}{\partial a_t^\beta} D_{t+1}^0 .
 \end{equation}
   Using Lemma 3 to tell us that $ D_{T-1}^{\alpha} > D_{T-1}^0$, one
may conclude from equation (\ref{e57}) that for $t = T-2$
 \begin{equation} \label{e58} 
 -\frac{\partial g}{\partial a_t^{\alpha}} < \frac{\partial
f}{\partial a_t^\beta}.
 \end{equation}
  By assumption (\ref{e5}), $ -\frac{\partial g}{\partial a_t^{\alpha}}
= -\frac{\partial f}{\partial b_t^\beta}$, so equation (\ref{e58})
implies
 \begin{equation} \label{e58a}
  -\frac{\partial f}{\partial b_t^\beta} < \frac{\partial f}{\partial
a_t^\beta}.
 \end{equation}

By assumption (\ref{e4}), if $ a_t^\beta = b_t^\beta$ then expression
(\ref{e58a}) would be an equality.  Since it is an inequality, the
concavity of $f$ in $a$ combined with assumption (\ref{e5}) tells us
that $ a_t^\beta < b_t^\beta$, and since in equilibrium $ b_t^\beta=
a_t^{\alpha}$, (\ref{e50}) is proved for the special case of $t=T-2$.

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

\bigskip
\noindent
 {\bf Part IV: Inequality (\ref{e50}) for General t.}\\
  What was special about $t=T-2$ in Part III was the use of Lemma 3.
One may prove inequality (\ref{e50}) for $t = T-3$ after first
proving a version of Lemma 3 for $V_{T-2}^{\alpha}$ instead of
$V_{T-1}^{\alpha}$. Let us use our new knowledge that inequality
(\ref{e50}) holds for $t = T-2$.  A generalized version of Lemma 3 is
    \begin{equation}\label{e62}
 D_t^{\alpha} > D_t^0. 
  \end{equation}

 The proof of Lemma 3 did not rely on many special characteristics of
the last two periods. The numbers ``T-1'' and ``T'' can everywhere be
replaced by ``t'' and ``t+1'', and the only difficulty is that the
proof did cite Proposition 2 and equation (\ref{e37}), which are
special to periods $T$ and $T-1$. 

   Proposition 2 was used to confirm that $ X_2$ and $X_3$ were
positive, which in the $T$-period game is equivalent to showing that
$a_{T-1}^0 - a_{T-1}^{\alpha} > 0$ and $ a_{T-1}^0 - a_{T-1}^\beta> 0.$
It was already shown that equation (\ref{e50}) holds for $t = T-2$,
so that can be used to prove that (\ref{e62}) holds for $t = T-2$.
The general form of (\ref{e37}) is 
   \begin{equation}\label{e63}
 D_{t+1}^{\alpha} > VD{t+1}^0.
  \end{equation}

  Lemma 3 itself replaces (\ref{e37}) for the case of $t = T-2$.
Hence, inequality (\ref{e62}) can be proved for $t = T-2$ using the
proof of Lemma 3 with two differences: instead of citing Lemma 1,
cite the part of Proposition 3 that was already proved; and instead
of citing (\ref{e37}), cite Lemma 3.

 Once it is shown that (\ref{e62}) holds for $t = T-2$, one may go
back and follow the outline of Part III of this proof to prove
inequality (\ref{e50}) for $t = T-3.$ After that is done, one can
prove that (\ref{e62}) holds for $t = T-3$, go back to Part III, and
continue recursively, which proves that (\ref{e50}) holds for any
value of $t$.\\
   Q.E.D.

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



\newpage


\begin{center}
{\bf References.}\\
\end{center}
 
 Aoki, Reiko (1988) ``R\&D Rivalry Over Time: A Dynamic Stochastic
Game Approach,'' mimeo, Dept of Economics, Ohio State University,
August 1988.

 
 Aron, D. and Edward Lazear. ``Competition, Relativism, and Market
Choice.''  Working Paper No. E-87-56. Stanford: Hoover Institution.
December 1987.\footnote{xxx  pubilished?}

 
 Beath, John, Yannis Katsoulacos and David Ulph.  ``Sequential
Product Innovation and Industry Evolution.'' {\it Economic Journal},
97 (Conference 1987): 32-43. 

 
 Beath, John, Yannis Katsoulacos and David Ulph (1989) ``Strategic
R\&D Policy,'' {\it Economic Journal}, 99 (supp): 74-83.


 
 Delabola 1988


 
 Dixit, Avinash. ``Strategic Behavior in Contests.'' {\it American
Economic Review}, 77 (December 1987): 891-898.
 
 Fudenberg, Drew, Richard Gilbert, Joseph Stiglitz, and Jean Tirole.
``Preemption, Leapfrogging, and Competition in Patent Races.'' {\it
European Economic Review }  22 (June, 1983): 3-35 .

 
 Fudenberg, Drew and Eric Maskin. ``The Folk Theorem in Repeated
Games with Discounting or with Incomplete Information.'' {\it
Econometrica} 54 (May 1986): 533-554.

 
 Gilbert, Richard and David Newbery.``Patenting and the Persistence
of Monopoly.'' {\it American Economic Review } 72 (June 1982):
514-526.

 
 Gilligan, Thomas, William Marshall, and Barry Weingast. ``Regulation
and the Theory of Legislative Choice: The Interstate Commerce Act of
1887.'' {\it Journal of Law and Economics}, forthcoming.
\footnote{xxx  pubilished?}

 
 Grossman, Gene. and Carl Shapiro. ``Dynamic R\&D Competition.'' {\it
Economic Journal} 97 (June 1987): 372-387.

 
 Harris, C. (1985). ``Dynamic Competition for Market Share: An
Undiscounted Model,'' mimeo, Nuffield College, Oxford, November 1988.

 
 Harris, C. and J. Vickers (1985), ``Perfect Equilibrium in a Model
of a Race,'' {\it Review of Economic Studies} 52 (April 1985):
193-209.

 
Harris, C. and John Vickers (1987) Racing with Uncertainty'' {\it
Review of Economic Studies}, 54, 1-21.

 
 Hirshleifer, Jack. ``The Analytics of Continuing Conflict.'' Working
Paper \#467A. Los Angeles: UCLA Dept of Economics. August 1987.
\footnote{xxx  pubilished?}

 
 Lazear, Edward \& Sherwin Rosen. ``Rank-Order Tournaments as Optimum
Labor Contracts.''  {\it Journal of Political Economy} 89 (October
1981): 841-864.


 
 Lippman, Steven and Kevin McCardle. ``Dropout Behavior in R\& D
Races with Learning.'' {\it Rand Journal of Economics} 18 (Summer
1987): 287-295.

 
 Nalebuff, Barry \& Joseph Stiglitz.  ``Prizes and Incentives:
Towards a General Theory of Compensation and Competition.''  {\it
Bell Journal of Economics.} 14 (Spring 1983): 21-43. 

  
 Reinganum, Jennifer. ``Innovation and Industry Evolution.''  {\it
Quarterly Journal of Economics} 100 (February 1985): 81-100.

 
 Vickers, John (1986) ``The Evolution of Industry Structure when
there is a Sequence of Innovations,'' {\it Journal of Industrial
Economics}, 35: 1-12.


 \end{document}
