Many economic problems can be modeled as n-person non-zero sum games. In such situations, there are gains to be had by coordination of strategies. Assuming there are no restrictions on side-payments, the players then bargain over the division of the gains. This note establishes that, for a restricted class of economic problems, the threat equilibrium in the bargaining game coincides with the perfectly competitive equilibrium

Reinganum, Jennifer F.

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DIVISION OF THE HUMANITIES AND SOCIAL SCIENICES 
CALIFORNIA INSTITUTE OF TECHNOLOGY 
PASADENA, CALIFORNIA 91125 
A NOTE ON RATIONAL THREATS AND COMPETITIVE EQUILiaRIUM 
Jennifer F. Reinganum 
.:.. � � 
ll,.� - - .. 1 ,, Slf�LL tA"" 
SOCIAL SCIENCE WORKING PAPER i� 
Januarj. 198 
Revised J unel 19 8 
A NOTE ON RATIONAL THREATS AND COMPETITIVE EQUILIBRI 
Jennifer F. Reinganum 
California Institute of Technology 
ABSTRACT 
Many economic problems can be modeled as n-person 
sum games. In such situations there are gains to be had lby 
coordination of strategies. Assuming there are no restr:ilcti'ons on 
side payments, the players then bargain over the divisio1 off t�e 
gains. This note establishes that, for a restricted clais o'f 
economic problems, the threat equilibrium in the bargain�ng !game 
coincides with the perfectly competitive equilibrium. 
A NOTE ON RATIONAL THREATS AND COMPETITIVE EQUILIBRIUM 
Jennifer F. Reinganum 
California Institute of Technology 
I. Introduction 
In economic analysis we are of ten concerned with problems 
which can be modeled as n-person, nonzero-sum (NPNZS) games. For example, 
there are oligopoly games, games of research and development, and 
games of resource extraction (caused by common property externalities) . 
In such situations it is possible to generate greater joint profits 
by coordinating the choice of strategies rather than playing 
noncooperatively. If there are no restrictions on side payments, 
we would expect to see such strategy cpordination. 
Consider a situation in which there are gains to cooperation 
and the parties agree to maximize joint profits and then to submit 
to arbitration to divide the profits. They know that the arbitrator 
will split the profits using the Nash Bargaining Solution (NBS) [Nash, 
1953]. In the NBS, the division of profits depends upon some reference 
point. That is, for the game with side payments, the NBS is 
<Pi 
V0 + (n - l) Vi l: vm 
________ ..!!!m.'fi 
n i 1,2, . . .  ,n 
where V0 =maximized joint profits and (v1,v
2
, ... ,Vn) is 
disagreement or reference point, an allocation generated 
noncooperative play. 
Nash (1953) , and Ha.rsanyi (1959) have argued 
the players intend to maximize joint profits and are 
over the split, it is reasonable 
in determining the reference point. That is, player i 
to choose a (threat) strategy si so as to 
.Thus the original game 
I will refer to the Nash equilibrium 
as a "rational threats equilibrium. " 
Thus the players report their 
they will do should negotiations break down) to the 
The arbitrator then calculates the reference 
the NBS. I am interested in the structure of these 
threats. 
Since static games are degenerate versions 
games, I will focus on differential games or games played 
time. 1 
3 
II. The Proposition 
In this note, I wish to focus on n-person games with a 
particular structure. Let x denote an n-tuple of state variables and 
s an n-tuple of controls. Suppose the players have identical strategy 
spaces and symmetric payoffs. To formalize this notion of symmetry, 
let the players' index set be I= {1,2, ... ,n}. Now define n: I+ I 
to be a permutation of the index set I. There are n! such permutations, 
indexed by k EK= {1,2, . • .  ,nl}. Thus x denotes the vector of state 1fk 
variables with indices permuted according to the k
th 
permutation 1fk. 
Define the function g = g (t,x,s) where 
g (t,x ,s ) 
1fk 1fk 
for all k and k' in K. 
g (t,x
1fk' 
,s
1fk'
That is, the function g looks the same for all permutations 
of the index set I. 
Similarly, define fi f(t,x,si) where 
f (t,x ,s.) 
1fk J. 
f(t,x ,s.) 
1fk I J. 
for all k and k' in K. 
Finally let hi = h (t,si,s/i) where s/i 
s. ) i+l' • • •  ,sn 
• Suppose that 
h (  t ,s., (s/i) ) J. 1fk 
h (t,s., (s/i) ) J. 1fk l 
(sl, ... ,si-1' 
for all k and k' EK. Assume that g, fi and hi are 
t and continuously differentiable in (x,s) . Then we 
payoff structure to be of the following form: V i, 
. 
J
T 
Vi (s) = [gs. - f .]dt 
0 
i i 
where xi= hi + o (t) xi' xi (O) = XO' and o (t) is
The corresponding threat game payoffs are 
<pi (s) 
V0/n + (l/n) JT[ (n - l) {gs. - f.} 
0 J. J. 
If we define the strategy space for i to be the set 
strategies (or piecewise continuous functions si (t) 
we can apply optimal control theory. 
Hi 
(n - 1) (g Si n - f.] - l/n E [g L - f ] + i m#i m m 
n 
E 
where Ai are time-dependent Lagrange multipliers and m 
jointly concave in (x,si) .• 
Each player i chooses si (t) so as to 
sj (t) as given for all j # i. When an equilibrium 
it satisfies the usual conditions:2 
ClHi (n -
Clsi 
= --n-
1) r� Cls i 
n 
+ E 
m=l 
Si + g 
- -2: - l/n E 
-
Clf.J Clg Clsi m#i·Clsi 
. Clh. 
Ai__!!!.= O in Clsi 
in 
the 
(1) 
5 
'i aa, l(n - 1) ['' ''<] ['' "m ]A = - - = - - • s - - - l/n 2: - · s - -j ax. n ax. i ax. 1'. ax. m ax. J J J mi J J 
+ A�o(t)1, A�(T) = 0 J J (3) 
Since the players are entirely symmetric, it is reasonable 
to look for a symmetric threat equilibrium. Then s� 1 
all i and j, and (2)-(3) reduce to 
(n - 1) [ 
n g - _2:. + 2: A� _2!!:. = O 
Clf. J n . Clh asi m=l m Clsi 
�� = - A�o(t), A�(T) = 0 J J J 
s� for J 
(4) 
(5) 
But (5) means that A�(t) = c�exp{- J
t 
o(T)dT}. Since
exp{•} is never zero, 3 A� (T) J 
for all i,j. 
0 implies that the constant c� J 
Thus we are left with the equilibrium condition 
g 
Clfi =as-• i 
i 1,2, ... ,n
0 
(6) 
In a variety of settings, (6) can be interpreted as a 
first-order condition for an instantaneously competitive equilibrium. 
That is, when the problem has the structure as summarized in (1) 
the threat equilibrium coincides with the instantaneously 
competitive (not merely Nash-Cournot) equilibrium. 
III. Examples 
Several examples may illustrate the nature 
tions, the types of economic problems which satisfy 
interpretation of condition (6) . 
1. 
Consider an intertemporal game of oligopoly 
goods. Then the price obtainable at t for a 
qi depends upon i's current output, everyone else's 
and the existing stock of the durable good. 
assume that there is no resale market), current demand 
the greater is the stock outstanding, since durable 
purchased only periodically. Thus the current price 
and we assume that the evolution of the stock Q is 
Q 2:q. - o(t)Q J 
where o(t) is the rate at which the stock Q 
or is consumed. 
If the production costs are c(t,qi) the 
Vi(q) J
T -rt 
-
0 
e [p(t,Q,2:qj)qi - c(t,qi)]dt 
where Q 2:q. - o(t)Q, Q(O) = Q . J 0 
In light of the foregoing analysis, the threat 
is (q!, ... ,q�) such that 
6 
p(t,Q,l:qf:) J 
<lc(t,q�) 
()qi 
i l, ... ,n 
But these characterize instantaneously competitive 
7 
behavior (price = marginal cost). So n identical oligopolists, 
in bargaining over the division of jointly maximized profits, will 
specify as their threat (or default) strategies the (instantaneously) 
competitive equilibrium strategies. 
2. 
Consider the game of research and development discussed 
in Reinganum (1979). The payoffs are 
Vi(ul'. • .  ,un) 
fT -A.l:z. e J [PA.u. 
0 J_ 
where zi = ui' zi(O) = z0, i = 1,2, . • .  ,n. 
-rt 2 e u/2ldt 
This game was formulated from the following scenario: 
if no firm has succeeded in completing the innovation by t (which 
-A.l:z 
occurs with probability e j), then each firm invests at the 
(discounted) rate $ e-rtu�/2 to increase its knowledge stock by u . •  
J_ J_ 
In addition, if no firm has yet succeeded, there is an instantaneous 
conditional probability of success of A.ui. Success is rewarded by a 
patent with value $P. 
The structure of this game is clearly subsumed by the 
conditions in (1). Thus at the rational threats equilibrium, 
ut = PA.ert 
8 
But these are precisely the competitive equilibrium stratlegies I derived (as the limit of an n-person Nash equilibrium as µ-><x>) 
Reinganum (1979). 
3. 
Suppose there is a pool of oil underlying the p 
n people (players). If they sell the oil competitively 
listically--see example 1) at price p per barrel and if e 
costs depend upon the number of barrels extracted ni and 
remaining reserves X(t), then we can formulate the payoff 
:PI:f:o:: 
trlc ion 
I pon e 
I as 
Vi(nl' ... ,nn) 
f T -rt e [pn. - c(t,X,n.)]dt
0 J_ 
J_ 
where x = -l:n . . J 
Suppose that lim �c (t,X,n.)
X-rO ni i 
never be completely exhausted. 
oo so that the r�sour 
Then we need not explicit�y lo
X(t) > 0. 
Again, at a rational threats equilibrium 
- <le P - a---n (t,x,nf:) i J_ 
i 1,2, ... ,n
e wi 
This coincides with myopic, instantaneously competitive b!ehavidL 
9 
IV. Conclusions 
Although the restrictions in (1) seem fairly strong, 
several interesting economic problems fit comfortably within the 
bounds. It is not clear that the restrictions (1) are necessary--
they are merely sufficient to generate the result that the threat 
equilibrium coincides with the (instantaneously) competitiv.e ·equilibrium. 
Recall that these threat strategies are never actually 
meant to be played--they are merely threats advanced during the 
bargaining process. But it is interesting to note that, for 
example, oligopolists will--in an effo.rt to obtain a greater share 
of the joint profits -- threaten to behave competitively. If 
bargaining breaks down for some reason, then -- because the threats 
are assumed to be binding -- the cartel will degenerate to perfectly 
competitive (not merely Nash-Cournot) behavior until bargaining 
can be successfully completed. 
1. 
FOOTNOTES 
The reader should be cautioned that I am merely s�eking 
::r. 
I repe 
nature of the threats which determine the disagre 
in a game which involves explicit cooperation and 
This ought not to be confused with the analysis o 
the 
oint 
games (or "supergames") as in Friedman (1971) , wh reih 
optima are supported by essentially noncooperativ plly
strategies of the form "I will play (single perio ) cbaiferatH�ly 
so long as you do; if you ever deviate, I will pl y (bi: gle
period) noncooperatively forever" are allowed. 
ion 
The games described in this note are not rep1atetl 
they are "one shot" games which� depend-: the evolu 
of an economic variable (or stock) through time. Thub 
result has nothing to do with (and claims no appl�cabll�ty 
his 
the repeated games problem. 
2. Given s , m f i, the problem is an n-state, 1 conlrol[ P m 
for player i. Since Hi is jointly concave in (x, i), 
the first-order conditions are necessary and suff�ciehtlto 
determine a maximizing control (if one exists). 
3.  For infinite horizon problems, -Jto(T)dT may appJoach �00 as t � oo. Thus the analogous transversality conditiol 
lim A(t) = 0 provides no info�mation. To deal wiJh tbi
t->OO 
difficulty, we define the infinite horizon game to be the 
limit of a sequence of finite horizon games as the horizon 
grows without bound. Then the solution to the infinite horizon 
game is defined to be the limit of the finite horizon solution 
as T + 00 (if such exists). This procedure is appropriate given 
certain growth restrictions (see Seierstad, 1977). 
11 
REFEREMCES 
Friedman, Avner, "Differential Games," New York: 
Sons, 1971. 
Friedman, James, A noncooperative equilibrium 
Review of Economic Studies 38 (1971):1-12 
Harsanyi, John C., A bargaining model for the 
game. In "Contributions to the Theory of 
by A. W. Tucker and R. D. Lucas, Princeton, 
Princeton University Press, 1959. 
Nash, John F. , Jr., Two-person cooperative games, 
21 (1953):128-140. 
Reinganum, Jennifer F., Dynamic Games of R 
Dissertation, Northwestern University, 
Seierstad, Atle, Transversality conditions 
with infinite horizons, Institut� of 
of Oslo, Norway, January 1977. 


A Note on Rational Threats and Competitive Equilibrium

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