Aggregative Games with Entry
Simon P Anderson, University of Virginia
Nisvan Erkal, University of Melbourne
Daniel Piccinin, Brick Court Chambers
Aggregative Games Workshop
University of Strathclyde
April 2011
Our Goal
• Explore long-run impact (i.e. with entry and exit) of
exogenous shocks in aggregative games
– Cost shocks (subsidies, tariffs)
– Changes in the objective function (cooperation,
privatization)
– Changes in timing of moves (simultaneous-move vs.
leader/follower)
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Preliminaries
• Two-stage game:
– In the first stage, players simultaneously make entry
decisions. Entry involves a sunk cost.
– In the second stage, after observing how many players
have sunk the entry cost, the players simultaneously
choose their actions.
• Second stage: A is the set of active players.
qi
• Aggregator: Q
i A
• Payoffs: πi (Q-i +qi, qi) or πi (Q, qi)
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Assumptions
Assumption 1 (Competitive Games):
πi (Q-i +qi, qi) is strictly decreasing in Q-i for qi > 0.
• Agents impose negative externalities upon each other.
• Rules out public goods contribution game.
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Assumptions
Assumption 2 (Payoff functions)
a) For any given Q i
, πi (Q-i +qi, qi) is continuous, twice
differentiable, and strictly quasi-concave in qi for Q i Q i ,
with a strictly negative second derivative with respect to qi at
the maximum.
b)For any given Q i
πi (Q, qi) is continuous, twice
differentiable, and strictly quasi-concave in qi for Q Q
with a strictly negative second derivative with respect to qi at
the maximum.
• 2(a) takes as given the actions of all others - relevant when
players choose their actions after entry.
• 2(b) takes as given the aggregator - relevant when one player
commits to its action before entry.
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Assumptions
We consider both strategic substitutes and strategic
complements
Assumption 3 (Reaction function slope)
d2 i
dqi2
d2 i
dqi dQ i
Gives us Lemma 1:
Q-i + ri (Q-i) is monotonically increasing in Q-i and so there
can be no over-reactions
Assumptions apply in wide range of settings:
– Bertrand (CES, logit) and Cournot oligopoly games
– R&D contests
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Cumulative best reply function
• ~ri (Q ) is the optimal action of firm i which is consistent
with a given value of the aggregator, Q (Selten, 1970).
Lemma 2:
d~
ri
dQ
ri '
1
1 r'
• Strategic substitutability or complementarity is
preserved in the cumulative best replies.
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Maximised Profit Function
• Define ~i* (Q) as the value of profit given a total output
Q and given that firm i maximises its profits given the
output of others.
Lemma 3:
~ * (Q )
i
is strictly decreasing for Q Q i
~
d
ri (Q)
d ~* (Q)
~
~
Proof:
i ,1 (Q, ri (Q))
i , 2 (Q, ri (Q))
dQ
dQ
d~
ri (Q)
d ~* (Q)
FOC gives us:
(1
) i ,1 (Q, ~
ri (Q))
dQ
dQ
Which is negative by Assumption 1 and Lemma 2
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ZPHEE
• Homogeneous entrants (E) and others (AI).
• Exogenous change (lower cost, superior product quality,
change in timing or objective function, etc.) affects some or all
of the firms in AI.
• Interested in equilibria such that AI A and marginal entrant
type is the same both before and after the exogenous change.
Definition: At a Zero Profit Homogeneous Entrants Equilibrium
(ZPHEE):
where E
~ * (Q )
i
A
K
for all i
~ * (Q )
i
A
K
for all
AI
A
i E
A
A is non-empty.
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Core Results: Positive
Suppose that there is some change affecting the firms in
AI , which shifts the ZPHEE set of agents from A to A’, both
of which contain AI and at least one fringe firm. Then:
• QA = QA’ (underpins several results in the literature)
• qi,A = qi,A’ for all i in E, and qi,A = qi,A’ for all unaffected firms in
AI.
• Any change making the affected firms more aggressive in
r (Q )s) will
aggregate (in the sense of raising the sum of their ~
decrease the number of fringe firms.
All of this even though the number of active firms may
change significantly.
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Core Results: Normative
• Change in producer surplus equals the change in rents
to the affected firms.
• Consumer surplus remains unchanged if it depends
solely on the value of the aggregator.
– More detail on this in a moment
• Hence, change in welfare is measured solely by the
change in the affected firms’ rents.
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Consumer surplus and Aggregative
Bertrand Oligopoly Games:
The “If IIA ...” Result
Let demand be quasi-linear and satisfy the IIA property (i.e.
Di/ Dj is independent of pk). E.g. CES/Logit
• Then applying Goldman
~ & Uzawa (1964) indirect utility has
the form: V ( p, Y )
[ gi ( pi )] Y
i
• So demand is given by: Di
~
'[
gi ( pi )]gi ' ( pi )
i
• Let qi=gi(pi)
• So we have an AG where consumer surplus depends only
on the aggregator
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Consumer surplus and Aggregative
Bertrand Oligopoly Games:
The “Only If IIA ...” Result

Assume indirect utility is given by: V ( p, Y )
~
[
qi ( pi )] Y
i

Then demand is given by Di ( p)
Then:
~
'[
qi ( pi )]qi ' ( pi ) 0
i
– we have an AG where consumer surplus depends only on
Q; and
– Demand satisfies the IIA property
Note: it is possible to construct Bertrand AGs that do not
satisfy IIA and therefore for which CS depends on more
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than just Q
Applications: Cooperation
• Suppose two firms (j and k) cooperate and a ZPHEE
prevails after the cooperation as well as before.
• Before the pact, firm j’s cumulative best reply is
defined by the solution to the FOC:
j ,1
(Q, q j )
j ,2
(Q, q j )
0
• After the pact, firms j and k maximise joint profits and
so qj is determined by:
0
j ,1 (Q, q j )
j , 2 (Q, q j )
k ,1 (Q, qk )
• QA, outsiders’ profits, and consumer welfare remain
unchanged.
• Payoffs to cooperating players all weakly decrease.
• This analysis implies that mergers are desirable in the
LR if the merging parties would like to do it.
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Applications: Leadership
• In a simultaneous move game, firm i’s cumulative best
reply is defined by:
~
(
Q
,
ri (Q))
i ,1
~
(
Q
,
ri (Q ))
i,2
0
• A Stackelberg leader in the LR rationally anticipates that
Q is independent of its output so its optimal choice is
defined by:
(Q, q ) 0
i,2
L
• Hence, by Assumption 2b the leader’s LR output must be
higher than in the simultaneous move game.
• That implies:
– Fewer fringe firms in equilibrium, although per-firm fringe
output is unchanged
– Total surplus is higher although consumer surplus is
unchanged
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j,2
(Q, q j )
q Cj : Cooperative action
q SM
j : Simultaneous-move
action
q Lj : Leader’s action
q SM
j
q Cj
j ,1
j,2
k ,1
0
j ,1
qj
q Lj
j,2
0
j,2
0
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Heterogeneous fringe: setup
• Crucial for the results above that the type of the
marginal entrant does not change.
• Suppose now that all fringe firms have the same profit
functions, except that they differ by idiosyncratic K. Let
entry costs be strictly increasing across fringe firms.
• Suppose one of the insider firms, j, experiences a
change that increases its marginal profit.
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Heterogeneous fringe: results
• QA < QA’ (irrespective of whether the actions of the firms are
strategic substitutes or complements)
• The change causes some fringe firms to exit (i.e., the number
of fringe firms is higher at A than at A’);
• The fringe firms which remain active in the market must be
earning lower rents after the change;
• Each fringe firm chooses a higher (lower) action if and only if
actions are strategic complements (substitutes); and
• Firm j chooses a higher action.
• Intuition for aggregator result: relies on Lemma 3 - since the
marginal entrant in A’ has lower entry cost than marginal
entrant in A, the marginal entrant in A’ must have lower
profits after the change. This implies that competitive
conditions (summarised by Q) must have become more
intense.
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Concluding remarks: Overview
• In AGs where the type of the marginal entrant does not
change, following an exogenous change:
– Even though the affected players’ equilibrium actions and
payoffs, and the number of active players change, the
aggregator stays the same: Free entry completely undoes the
short-run impact
– Unaffected players’ equilibrium payoffs remain unchanged,
whether or not actions are strategic complements or strategic
substitutes.
– Welfare results follow if welfare depends on the aggregator
only: Cournot, IIA - CES/Logit.
• Results apply to a wide range of problems with (i) an AG
structure and (ii) entry:
– Cost changes, mergers, privatisation, RJVs, R&D contests,
strategic investment, international trade, etc.
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Contribution to the AG literature
• We add free entry:
– Allows us to model effects of exogenous changes on market
structure (i.e. to capture full, long run effects)
• We show the connection between the IIA property and
the aggregative structure of consumer welfare for
Bertrand Oligopoly Games
– This result allows us to generate normative results from AG
analysis for Aggregative Bertrand Oligopoly Games
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Contribution to applications literatures
• Identify the underlying reason for a wide range of
results:
– These results depend critically on choice of AG structure
and are actually just special cases of a more general result
• Generalise results and show their limits
– E.g. homogeneous product Cournot results (mergers,
trade) extend to differentiated IIA Bertrand mergers
– E.g. normative results from leadership analysis only extend
to homogeneous Cournot and IIA Bertrand systems
• Results derived from the common framework can be
meaningfully and easily extended
– E.g. What if privatisation takes place by merger with an
existing private competitor, and/or results in improved
efficiency? No change in result that consumer welfare is
unaffected
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Concluding Remarks: Caveats
• Integer constraints
– Can at least put bounds on changes to aggregator
• Localised competition
– Two firm Hotelling and three firm circular city model are
aggregative in the short run (i.e. without entry) but not in
the long run
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