Well-behaved [and Smooth]
Aggregative Games
Richard Cornes [ANU]
Roger Hartley [Manchester Univ]
What is an Aggregative Game?
An aggregative game is one in which each player’s payoff function
has the form


 i x    i xi ,  j 1 x j   i xi , X .
n
Examples include:
Cournot oligopoly/oligopsony
Pure public good provision
Open access resources [Tragedy of the commons]
Contest theory
Diamond’s coconut search model
Tragedy of the anticommons
Aggregative Games: Individual Behavior
A player’s marginal payoff function can be written as
 i xi , X   i xi , X 
 i xi , X  

xi
X
Ignoring corner solutions – which are easily accommodated the FOC associated with player i’s best response requires that
 i xˆi , X   0
Her replacement function [more generally, correspondence] is
xˆi  ri  X 
A Well-known Example: Cournot
Inverse demand function: P X   a  bX
•
• Cost Functions: Ci(xi) = cixi
vi xi , X   a  bX xi  ci xi
  i xˆi , X   P X   xˆi P'  X   ci  0
• In explicit form, taking account of nonnegativity constraints:
 a  ci  bX 
xˆi  ri  X   max
,0,
b


defined for ri  X   X .
Linear Cournot Geometry I
qi
qˆi  ri Q
qˆi  bi Qi 
Qi 
n
q
j 1, j  i
j
Q
Aggregative Games: Equilibrium
• A Nash noncooperative equilibrium of an n-player game is
conventionally described by the n equations provided by the n
players’ best response functions. This is problematic.
• Using replacement functions, a Nash equilibrium is described
by one equation in one unknown – the aggregative
replacement function - regardless of how many
heterogeneous players there are:
n
R X    r j  X   X .
j 1
• An alternative useful description is obtained by dividing both
sides by the quantity X:
 n rj  X   n

   s j X   1


X
j

1

 j 1
Well-behaved Games
• We are interested in games that are wellbehaved – that is, games that possess a
unique equilibrium in pure strategies.
• Specifically, we will present conditions that
– are sufficient for a game to be well-behaved,
– are easy to check,
– have a straightforward geometric intuition,
– are less restrictive than any with which we are
familiar
Familiar [and Restrictive] Conditions
• Suppose we assume
 i xi , X 
C 1:
 0, 0  xi  X
xi
and
 i xi , X 
C2 :
 0, 0  xi  X
X
These conditions imply that
xˆi
 i xi , X  X

 0,
X
 i xi , X  xi
which implies a downward-sloping replacement function
for player i.
Hence the aggregate replacement function is downward-sloping
wherever it is strictly positive. This establishes uniqueness.
Existence is easily established from continuity considerations.
Digression into best response functions
 i xi , X   0 beforeand afteran exogenousshock
 i xi , X 
 i xi , X 


xi
dxi 
X
dxi  dX i   0
 i  xi , X 
dxi
X

 i  xi , X   i xi , X 
dX i

xi
X
dxi
 1 
0
dX i
These are the familiar bounds on the slopes of best response functions.
It’s obvious why they work when n=2.
Less obvious, though, when n>2!
Linear Cournot Geometry II
ri(X)
r1(X)+r2(X)+r3(X)
EN
r1(X)+r2(X)+r3(X)=X
r3(X)
r2(X)
r1(X)
45o
X
The Familiar Conditions and Cournot
In Cournot with possibly nonlinear demand and cost
functions,
vi xi , X   P X xi  Ci xi    i xi , X   P X   Ci ' xi   xi P'  X 
so that
 i xi , X 
C 1:
 0  P'  X   Ci ' ' xi   0
xi
and
 i xi , X 
C 2:
 0  P'  X   xi P' '  X   0
X
C1 and C2 are stated by Hahn [RES 1962] as
sufficient for stability in oligopoly.
They are stated by Corchon more generally in
terms of the marginal payoff function as sufficient for
existence and uniqueness in general aggregative games.
Some Awkward Examples
• Consider Cournot again, but with a constant elasticity
demand function. This does not satisfy the familiar
condition. But as we will show, it may be perfectly
well-behaved.
• The same is true of even the simplest Tullock contest
• ... and of other examples, such as the static open
access resource model, and other cost- and surplussharing games.
Our Central Aim
We now present conditions, sufficient for a game
to be ‘well-behaved’, that are
•Less restrictive than Hahn’s conditions
•Naturally satisfied by all the awkward examples
•Capable of a natural behavioral interpretation
•As general as one can expect to find
Our Sufficient Condition for
‘Good Behavior’
• Our two conditions are:
A1: If xi , X  satisfies0  xi  X and  i xi , X   0 , then
 i xi , X 
0
xi
A2 : If xi , X  satisfies0  xi  X and  i xi , X   0 , then
 i xi , X 
 i xi , X 
xi
X
0
xi
X
The meaning of the new condition
• The familiar (Hahn/Corchon) conditions require the
replacement function, ri(X), to be everywhere
downward-sloping.
• Perhaps more familiar, they impose upper and lower
bounds on the negatively sloped best response
functions.
• Our (new) conditions do not require monotonicity of
the replacement function, but of the share function,
ri  X 
si  X  
X
• We call a player with a monotonic share function
‘regular’.
More Geometry
The following situation is not allowed by the familiar condition,
but is allowed by us:
xi
xi=ri(X)
ri(X*)
si(X*)
X
X*
The Share Function: A Simple Example
Cournot with unit elasticity: 4 firms, p = 1/X,
constant but idiosyncratic unit costs: (c1, c2, c3, c4) = (1, 2, 4, 5)
vi xi , X   p  X xi  ci xi
1
xi  ci xi
X
1
1
FOC :  i  xi , X    ci  2 xˆi  0
X
X
xˆi  ri ( X )  X  ci X 2
xˆi
 si ( X )  1  ci X
X

More correctly:
si(X) = max{1 – ciX, 0}
Our Conditions and Cournot
Suppose
(i) Inversedemandsat isfies p '  X   0 X  0,
Xp X   0 as X  0 and p  X   0 asX  .
(ii) For each i  I , we haveci '  xi   0 if xi  0 and
ci ' '  xi   maxp '  X ,2 p '  X   Xp' '  X if 0  xi  X .
Then our conditions A1 and A2 hold.
Our conditions permit some degree of diminishing costs.
For example, if demand is linear, diminishing costs do not threaten
existence of unique equilibrium as long as ci ‘’ > b.
Our Conditions in Other Models
• Simple Tullock contests violate the familiar conditions. But
they satisfy our less restrictive ones.
• Our approach can analyse more complicated Tullock
contests, allowing for risk aversion, more general contest
success functions, and heterogeneity, and can identify
conditions in these more general settings that are sufficient
for ‘good behaviour’.
•The simple over-fishing problem [tragedy of the
commons], too, does not satisfy the familiar conditions, but
does satisfy ours under standard assumptions.
•Other simple cost and output sharing games have an
aggregative structure, and under standard assumptions can
be shown to be well-behaved.
Beyond Well-behaved Games
• Our sufficient conditions for uniqueness should not be
regarded as sacred – many games violate them, and may have
multiple equilibria.
• Our methods – using replacement and share functions – can
[and should] be used to analyse any game that has suitably
aggregative structure.
• The important point to take from this paper is the idea of
conditioning player i’s behaviour on a common aggregate,
rather than on the sum of all other players’ choices, excluding
that of player i herself.
Badly Behaved Games: An Example
Tullock contest, identical risk averse players:
Suppose that u w  w  0.45w2 ,
 
w1
Then preferred share values are defined implicitly by
1
0.55  0.9Y   0
 1  0.9 1  Y  
Y
s(Y)
1
1
0.55 0.566
Y
Badly Behaved Games: Example (cont.)
• If each of 10 players has the utility function of our
example, then there are multiple equilibria:
– at the symmetric equilibrium, xi = 0.05633 for all i=1,…,10
– There are asymmetric equilibria in which three players
each have x = 0.05633, the remaining seven each have x =
0,
– There are asymmetric equilibria in which four players each
have x = 0.14, the remaining seven each have x = 0,
– And so on
Badly Behaved Games: Another
Example
si(X)
r 1
r
Tullock contest:
Identical risk neutral
players
pi 

xir
r
x
j 1 j
n
Where r > 1
X
Badly Behaved Games: Yet another
example
Diamond’s search model:
If you really want to rule out many equilibria in Diamond’s model, you can do so by
(i) making search costs rise very steeply with the level of search, and also
(ii) Assuming that, even if no-one else searches, player i will find an edible coconut
lying on the ground with prob > 0 if he searches.
Relaxing our Conditions: I
• We do not claim that our conditions are
necessary and sufficient.
• But we do claim that, if they are relaxed,
existence of a unique equilibrium may be
jeopardized unless a little extra structure is
imposed on the game.
Relaxing our Conditions: II
• Suppose there are nt identical
players of type t with a nonmonotonic share function,
• However, suppose that for
share values less than 1/nt,
their share functions are
regular.
• Then, the aggregation of their
share functions ensures that
the aggregate share function
is monotonic over the
relevant range.