M . Dolores Alepuz, A. Urbano Mathematical Social Sciences 37 1999 165 –188
167
2
too accurate by experimentation. Hence, the rival will be better informed and a tougher competitor: too precise signals discourage experimentation by duopolists.
The value of information in oligopoly games has been the subject of intensive research. These studies typically assume either that firms transmit information through
‘certifiable verifiable announcements’ or that the signals that yield information to the
3
firms are exogenously generated. In contrast with them, our model endogenously determines the amount of information available to firms. The closest model to ours is the
duopoly experimentation model of Mirman et al., 1994, who have also generalized the results of Mirman et al., 1993a. Their main contribution is to extend duopoly
experimental behavior to mixed strategy equilibrium and to show some cases for which the net value of information for the duopoly is positive. However, they are unable to
relate the duopoly experimental behavior with that of the monopoly in their general setting. Related to our model but with a different purpose are Aghion et al., 1993;
Alepuz and Urbano, 1993; Harrington, 1995. They assume that the degree of substitution between products is unknown to explain the phenomenon of price dispersion under
oligopoly. Finally, Alepuz and Urbano, 1997 deal with experimental behavior but in asymmetric heterogeneous duopoly markets.
The plan of the paper is as follows. The model and informational assumptions are laid down in Section 2. Then, we analyze the main features of information and the
experimental behavior of firms. Section 3 is devoted to show that larger outputs lead to more informative market signals. Monopoly and duopoly experimentation are compared
in Section 4. Some concluding remarks are given at the end.
2. The Cournot model
Consider a two period duopoly model. The firms produce a homogeneous product over the two periods. Inverse market demand in each period is given by
1 2
˜ ˜
˜ P 5 a 2 u q 1 q 1 ´
t t
t t
i
˜ where P is the price in period t, t 51, 2 and q , i 51,2, is firm i’s quantity in period t.
t t
˜ ˜
The parameter u is the fixed random slope and ´ is each period’s random demand
t
shock. It is assumed that each of these two random variables has full support on R and that they are independently and normally distributed. These distributions are known by
2
Note that here, in contrast to the duopoly models of information sharing, the key question is not whether firms are willing or not to share information but how much information to produce, since all the information
contained in the market signal is immediately shared with the rivals. See Malueg and Tsutsui, 1996 and references herein.
3
See Clarke, 1983; Gal-Or, 1985, 1986; Fried, 1984; Li, 1985; Logan, 1988; Novshek and Sonnenschein, 1982; Shapiro, 1986; Vives, 1984 among others.
168 M
. Dolores Alepuz, A. Urbano Mathematical Social Sciences 37 1999 165 –188
the firms. Firms’ market interaction is modeled as a game of imperfect information. The game unfolds as follows:
Description of the game. The game consists of two periods. In period one we distinguish two stages:
In stage 1, Nature chooses values u and ´ of the two independent random variables
1 2
˜ ˜
˜ u |Nm, h and ´ |N0, t respectively, where h51 s is u ’s precision and
1 u
2
˜ t 51 s is ´ ’s precision. As we noted above the Normal distributions of these two
´ 1
random variables are common knowledge, but not their realizations.
1 1
In stage 2, firms choose quantities q , q simultaneously and independently. These
1 2
1 2
quantities become commonly known. Let Q 5q 1q , then the price P is realized
1 1
1 1
where P 5 a 2 uQ 1 ´
1
1 1
1
Firms observe P but not u and ´ .
1 1
Period two consists of two similar stages: ˜
In stage 1, Nature selects a value ´ of the random variable ´ , which has the same
2 2
1 2
˜ distribution as that of ´ . This selection is independent of P , q and q .
1 1
1 1
1 2
In stage 2, firms choose quantities q , q simultaneously and independently which
2 2
1 1
are functions of P and Q . Let Q 5q 1q . Then the price P is determined by
1 1
1 2
2 2
P 5 a 2 uQ 1 ´ 2
2 2
2
which, as before, is observed by the firms. We assume that production cost are zero or that P is the net price received in t, after
t
subtracting a constant marginal cost that is the same for both firms, and that firms seek to maximize the sum of the two period expected profits.
Our analysis is confined to the pure strategy subgame perfect equilibria of this two period game. The equilibrium is calculated in the backward induction way. We begin by
1 2
analyzing the equilibrium of the second period as a function of P , q and q . After
1 1
1 1
2 1
2
ˆ ˆ
observing P , q and q , both players choose q , q such that:
1 1
1 2
2 i
i 1
2
ˆq 5 Argmax q E[P P , q , q ] i 5 1, 2 3
2 2
2 1
1 1
1 2
To solve this problem we have to find out first what is E[P P , q , q ] i.e.
2 1
1 1
1 2
˜ ˜
E[a 2uQ 1´ P , q , q ]. Since
2 1
1 1
1 2
1 2
˜ ˜
˜ E[a 2 uQ 1 ´ P , q , q ] 5 a 2 E[u P , q , q ]
4
2 1
1 1
1 1
1 1
2
˜ it is sufficient to find E[u P , q , q ]. By Eq. 1
1 1
1
˜ a 2 P
´
1
˜ ]]
] 5
u 2 5
Q Q
1 1
Therefore for each value of the parameter u, ˜
a 2 P
1 2
]] | Nu, tQ 6
1
Q
1
M . Dolores Alepuz, A. Urbano Mathematical Social Sciences 37 1999 165 –188
169
˜ The value of u is unknown to firms, but they a priori believe that u |Nm,h. Then,
after observing P and knowing Q , their new belief about u is that it is a realization of a
1 1
ˆ ˆ random variable which is Normally distributed according to Nu, h see DeGroot, 1970,
4
page 167, where a 2 P
1 2
]] mh 1 tQ
S D
1
Q mh 1 tQ a 2 P
1 1
1
ˆ ]]]]]]
]]]]] u 5
5 7
2 2
h 1 tQ h 1 tQ
1 1
and
2
ˆ h 5 h 1 tQ
8
1 1
2 1
2
˜ ˜
˜ Hence, E[u P , q , q ]5u and by Eq. 4, E[P P , q , q ]5a 2uQ . Inserting this
1 1
1 2
1 1
1 2
result in Eq. 3, we obtain the following unique symmetric solution: a
1 2
ˆ ˆ
ˆ ]
q 5 q 5 qu 5 9
2 2
ˆ 3u
Hence, both firms equilibrium expected profits in the second period are functions ˆ
Vu , where
2
a ˆ
ˆ ˆ
ˆ ]
Vu 5 qu a 2 2uqu 5 10
f g
ˆ 9u
Therefore, ˆ
Lemma 1. V. is a decreasing, strictly convex function of u.
The intuition of this lemma is clear. The first property of V. is that the higher the ˆ
value of u, the mean of the demand’s unknown slope, the smaller the second period ˜
expected profits to each firm. The convexity of V. means that information about u is ˜
valuable for each duopolist from an ex-ante viewpoint. Let u 5u and let Vu be each
5
˜ ˜
duopolist’s second period profits when u 5u is known. To see that information about u is valuable for firm i, it suffices to compare between firm i’s expectations over second
˜ period profits when the true value of u is to be learnt between periods one and two and
when it is not. In the former ‘informed’ case second period expected profits are equal to:
4
ˆ Note that given our normality assumptions, the signal and updated beliefs, u, may take on negative values.
Firms are constrained to choose positive prices and quantities. For convenience, we ignore this and, given the firms’ strategies that we derive, we can get negative prices and outputs for certain combinations of the signal
ˆ and u. The probability of such an event can be made arbitrarily small by appropriately choosing the variances
of the model.
5 2
i
Vu 5qu [a22uqu ]5a 9u , where q u is firm i’s profits maximizer quantity in period two and
2 1
2
where q u 5q u 5qu by symmetry.
2 2
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. Dolores Alepuz, A. Urbano Mathematical Social Sciences 37 1999 165 –188
1`
˜ E
f
Vu
g
5
E
Vu gu du
2`
˜ ˆ ˆ
where g is the density function of u which is Normally distributed according to Nu, h and in the latter ‘uninformed’ case second period expected profits are simply
˜ ˆ
V [Eu ]5Vu . ˜
Definition 1. Information about u is valuable for firm i, i 51, 2, if
˜ ˜
ˆ E[Vu ] . V [Eu ] 5 Vu
11
6
ˆ The strict convexity of V. in u guarantees, by Jensen’s Inequality, that Eq. 11 is
satisfied. 2.1. The analysis of the first period
Given the analysis of the second period, one can write each firm’s two period expected profits as a function of first period outputs. Denote first
1 2
i
˜ ˜
E[a 2 u q 1 q 1 ´ q ] i 5 1, 2 if P . 0
i 1
2 1
1 1
1
p q , q 5 12
H
1 1
1
otherwise to be the first period expected profits and let
i 1
2 i
1 2
ˆ P q , q 5 p q , q 1 E[Vu ]
13
1 1
i 1
1
be the two period profits to firm i. Observe that
1`
ˆ ˆ
E[Vu ] 5
E
VuP , Q fa 2 P dP 14
1 1
1 1
2`
˜ where f is the density function of the random variable a 2P . Since
1 1
2
˜ ˜
˜ ˜
˜ a 2 P 5 u q 1 q 1 ´ 5 uQ 1 ´ .
1 1
1 1
then, for each value of Q ,
1
ht ˜
]]] a 2 P | N mQ ,
. 15
1 1
2
S D
h 1 tQ
1 1
2
ˆ ˆ
The problem of the firms is to choose q and q , such that
1 1
6
An alternative explanation of the relationship between the convexity of V. and the value of information is the following: V. convex means that firms are risk-loving so that they prefer more variability in their posterior
˜ mean on u. This can be achieved by increasing the amount of new information concerning the slope of
demand. See Harrington, 1995.
M . Dolores Alepuz, A. Urbano Mathematical Social Sciences 37 1999 165 –188
171
1 1
2 2
1 2
ˆ ˆ
ˆ ˆ
q [ Argmax P q , q , q [ Argmax P q , q ,
1 1
1 1
1 1
1 1
16
1 2
q q
1 1
1 2
¯ ¯
Let q and q
be the first period myopic solutions. Namely,
1 1
1 1
1 2
2 2
1 2
] ]
] ]
q [ Argmax p q , q , q [ Argmax p q , q ,
1 1
1 1
1 1
17
1 2
q q
1 1
1 2
1 2
ˆ ˆ
¯ ¯
We are interested in comparing q , q , with q , q . Suppose that the conditions
1 1
1 1
7
for the existence of first period subgame perfect equilibria in pure strategies are fulfilled. Next, we characterize any selection of these equilibria.
1 2
¯ ¯
The first order conditions for the myopic solutions q , q set up current marginal
1 1
1 2
ˆ ˆ
profits equal to zero, meanwhile the first order conditions for q , q , are:
1 1
i i
j
ˆ ˆ
≠ p q , q
≠E[Vu ]
1 1
1
]]]] ]]]
1 5 0 i 5 1, 2, i ± j
18
i i
≠q ≠q
1 1
i
ˆ The first result of this paper is to show that E[Vu ] is an increasing function of q for
1
i 51, 2, which implies that each firm increases output with respect to the myopic choice for experimental purposes. Thus, lemma 2 describes the effect of the incentives to
manipulate information on the best response maps of the firms.
Lemma 2.
ˆ ≠E[Vu ]
]]] . 0 .
i
≠q
1
Proof. By Eq. 14:
ˆ ˆ
≠fa 2 P ≠E[Vu ]
≠ u
1
ˆ ˆ
]]] ]
]]] 5
E
V 9u fa 2 P 1 Vu
dP 19
F G
i i
1 i
1
≠q ≠q
≠q
1 1
1
In Appendix A we show that Eq. 19 can be expressed in the following way: ˆ
ˆ ≠E[Vu ]
1 ≠
u ˆ
]]] ]
] 5
E
V 0u 2
fa 2 P dP 20
F G
i 1
1
ˆ ≠P
≠q h
1 1
3 2
ˆ ˆ
ˆ By lemma 1, Vu is strictly convex. In fact V 0u 52a 9u . By Eq. 7,
ˆ tQ
≠ u
1
] ]]]
5 2 , 0 ,
2
≠P h 1 tQ
1 1
thus,
7 i
Notice that Eq. 13 is not, in general, quasi-concave in q , although it is continuous in it.
1
172 M
. Dolores Alepuz, A. Urbano Mathematical Social Sciences 37 1999 165 –188
2
ˆ 2a tQ
≠E[Vu ] 1
1
]]] ]]
] 5
E
fa 2 P dP . 0 21
i 1
1 2
3
ˆ ˆ
≠q 9h
u
1 i
j
where Q 5q 1q . j
1 1
1
Lemma 2 comports well with previous results in the literature. In particular, it is a generalization of the two values case in Mirman et al., 1993a to a duopoly setting. It is
also an alternative and a more general approach to Grossman, Kilhstrom and Mirman’s
8
paper who dealt with the n values’ case and with a normally distributed shock. A result similar to ours is obtained by Mirman et al., 1994 who also extended the model of
Mirman et al., 1993a to a duopoly market. However, since their analysis is carried out under the assumption of two values of the unknown parameter, our model can be
considered as well as a generalization of their results to the continuum case. The next proposition extends the results from best-replay mappings to equilibrium quantities. In
particular, it shows that at the equilibrium the Cournotian duopolists experiment by increasing quantity with respect to their myopic choice.
1 2
1 1
¯ ¯
ˆ ¯
Proposition 1. Let q , q be the myopic Cournot quantities . Then q .q
and
1 1
1 1
2 2
i
ˆ ¯
ˆ q . q . In particular
, q .0
1 1
1
Proof. By Eq. 18
ˆ a
1 1 ≠E[Vu ]
i j
ˆ ]
] ˆ ] ]]]
q 5 2
q 1 i 5 1, 2, i ± j
22
1 1
i
2m 2
2m ≠q
1 1
2
ˆ E[Vu ] depends on Q 5q 1q , hence
1 1
1
ˆ ˆ
≠E[Vu ] ≠E[Vu ]
]]] ]]]
5 .
1 2
≠q ≠q
1 1
1 2
ˆ ˆ
Then by Eq. 22 q 5 q , and by Eqs. 20 and 21,
1 1
2
ˆ 2a tQ
a 1
1 1
2
ˆ ˆ
] ]]
] q 5 q 5
1
E
fa 2 P dP 23
1 1
1 1
2 3
3m ˆ
ˆ 27mh
u
1 2
ˆ ˆ
ˆ where Q 5 q 1 q .
1 1
1
On the other hand, by solving Eq. 17 one shows that, a
1 2
] ]
] q 5q 5
24
1 1
3m
1 1
2 2
ˆ ¯
ˆ ¯
Consequently by Eq. 23; q . q , q . q , and
1 1
1 1
8
Grossman et al., 1977 GKM examine a consumer whose utility is given by z 1bx 1´ where ´ is random, z ]
and x are quantities of consumption goods, and b [ hb,bj. They show that the consumer increases ]
consumption of x to spread the mean utility functions further apart and hence increase the informativeness of realized utility.
M . Dolores Alepuz, A. Urbano Mathematical Social Sciences 37 1999 165 –188
173
2
ˆ 4a tQ
2a 1
2a ]
1
ˆ ]
]] ]
] Q 5
1
E
fa 2 P dP . 5Q j
1 1
1 1
2 3
3m ˆ
ˆ 3m
27h m u
A result similar to Proposition 1 was first obtained by Grossman et al., 1977. However, the basic idea behind the proof of our result is based on the notion of ‘more
informative signals’ while that of GKM is it on ‘sufficient experiments’. Nevertheless, by Blackwell, 1951 Theorem one can go from one direction to the other. The same
findings can be obtained under the Monotone Likelihood Ratio Property MLRP as in
ˆ Mirman et al., 1993a and Mirman et al., 1994. The intuitive explanation is that if Q .Q,
ˆ then the information structure induced by Q is sufficient for that induced by Q in the
sense of Blackwell, thus making price by Blackwell’s theorem a more informative signal. To the best of our knowledge, no paper in market experimentation has shown yet
directly that higher quantities produce more informative market signals. We are able to prove it since we deal with the closed form of the value function and the random
variables are normally distributed. This is the aim of the next section.
3. More informative signals
