Institute of Mathematics
Chair of Applied Mathematics II
Prof. Dr. C. Kanzow
Effort Maximization in Asymmetric n-person Contest
Games – An MPEC in Economy
Alexandra Schwartz (with J. Franke, C. Kanzow, and W. Leininger)
May 19th, 2011
www.mathematik.uni-wuerzburg.de/am2.html
Research partially supported by a grant from the Elite-Network of Bavaria
Outline
The Model
Lower Level Contest Game
Effort Maximization Problem
Special Cases
Analytical Solution
Lower Level Contest Game
Effort Maximization Problem
Results
Outlook
More General Utility Functions
Reformulation as MPCC
Alexandra Schwartz (with J. Franke, C. Kanzow, and W. Leininger)
2/17
The Lottery Model
Consider a lottery with n ≥ 2 participants, where
I xi ≥ 0 is the effort exerted by participant i,
I βi > 0 are his costs for exerting effort,
I he can win a prize of unit value,
I his winning probability is given by his relative effort Pnxi
x
j=1 j
.
Every participant then tries to maximize his utility function
xi
θi (xi , x−i ) = Pn
− βi xi
j=1 xj
depending not only on his effort xi but also on the effort of all other
participants x−i .
=⇒ Nash equilibrium problem
Alexandra Schwartz (with J. Franke, C. Kanzow, and W. Leininger)
3/17
Nash Equilibrium
Definition
A vector x ∗ is called a solution of the Nash equilibrium problem or a
Nash equilibrium, if for every player i = 1, . . . , n the strategy xi∗ ≥ 0 and
solves the problem
∗
max θi (xi , x−i
) subject to xi ∈ Xi .
xi
Theorem (Szidarovsky & Okuguchi 1997, Cornes & Hartley 2005)
The contest game posesses exactly one Nash equilibrium x ∗ for all β > 0.
Alexandra Schwartz (with J. Franke, C. Kanzow, and W. Leininger)
4/17
Effort Maximization Problem
Now assume that the lottery organizer wishes to maximize the
equilibrium effort excerted by all participants
θ0 (α) :=
n
X
xi∗ (α)
i=1
by manipulating the lottery as follows:
A weight αi > 0 is assigned to the effort of each participant such that his
winning propabilities now are Pnαi xαi x and consequently his utility
j=1
function is
j j
αi xi
− βi xi .
θi (xi , x−i ; α) = Pn
j=1 αj xj
Alexandra Schwartz (with J. Franke, C. Kanzow, and W. Leininger)
5/17
Problem & Applications
How should the lottery organizer choose the weights αi > 0 in order to
P
maximize the equilibrium effort ni=1 xi∗ (α)?
=⇒ mathematical problem with equilibrium constraints
Applications for this model are:
I
Lobbying
I
Public procurement
I
Affirmative action
I
Promotion tournaments
I
...
Alexandra Schwartz (with J. Franke, C. Kanzow, and W. Leininger)
6/17
Special Cases
Up to now, only the following two cases had been analyzed:
I
2-player case: (Nti 2004)
If n = 2, we have αi∗ = βi for i = 1, 2.
Thus, in this case the heterogeneity of the players is completely
removed.
I
homogeneous n-player case: (Dasgputa & Nti 1998)
If βi = β for all i = 1, . . . , n, we have αi∗ = 1 (or equivalentely
αi∗ = β) for all i.
Alexandra Schwartz (with J. Franke, C. Kanzow, and W. Leininger)
7/17
Nash Equilibrium of the Contest Game
Theorem (Corollary of Stein 2002)
For every choice of α > 0, the contest game hat exactly one Nash
equilibrium and the equilibrium efforts are given by
 "
#


k(α)−1
k(α)−1
β
1
i

P
βj
βj
αi 1 − αi P
∗
xi (α) =
j∈K (α) αj
j∈K (α) αj



0
if i ∈ K (α),
if i ∈
/ K (α),
where k(α) := |K (α)| and the set of active players K (α) is implicitly
given by




X
β
β
i
j
K (α) = i = 1, . . . , n (k(α) − 1) <
.

αi
α 
j∈K (α) j
Alexandra Schwartz (with J. Franke, C. Kanzow, and W. Leininger)
8/17
Effort Maximization Problem: Nonuniqueness of a
Solution
The formula for x ∗ (α) immediately implies the following two results:
Lemma
I
If α > 0 and c > 0, then K (α) = K (cα) and θ0 (α) = θ0 (cα).
I
If α, α0 > 0 with αi = αi0 for all i ∈ K (α) and


(k(α) − 1)βi 
αi0 ∈ 0, P
βj
j∈K (α) αj
for all i ∈
/ K (α), then K (α) = K (α0 ) and θ0 (α) = θ0 (α0 ).
=⇒ If there is a solution of the MPEC, it is not unique.
Alexandra Schwartz (with J. Franke, C. Kanzow, and W. Leininger)
9/17
Effort Maximization Problem: Objective Function
Replacing x ∗ (α) by the previous formula, we obtain
θ0 (α) =
n
X
xj∗ (α)
j=1

k(α) − 1  X
= P
βj
j∈K (α) αj
1
k(α) − 1
−P
βj
α
j∈K (α)
j∈K (α) j
αj

βj 
α2
j∈K (α) j
X
and thus have to solve the following problem:
max θ0 (α) subject to α > 0
α
Alexandra Schwartz (with J. Franke, C. Kanzow, and W. Leininger)
10/17
Effort Maximization Problem: Existence of a Solution
Problems:
I
Continuity of θ0 (α) is not guaranteed.
=⇒ θ0 (α) can be proven to be continuous on Rn++ .
I
The feasible set {α > 0} is unbounded.
=⇒ θ0 (α) is homogeneous in α, hence we can restrict ourselves to
P
those with nj=1 αj = 1.
I
The feasible set {α > 0} is open.
=⇒ It is possible to extend θ0 (α) continuosly on the closure of the
feasible set.
=⇒ The effort maximization problem posesses a (nonunique) solution.
Alexandra Schwartz (with J. Franke, C. Kanzow, and W. Leininger)
11/17
Effort Maximization Problem: Calculation of the Solution
I
The global maximum α∗ can be chosen such that θ0 (α) is smooth.
=⇒ α∗ is a stationary point, i.e. ∇α θ0 (α∗ ) = 0.
I
Every stationary point can be identified with a unique subset of the
players.
=⇒ There are only finitely many stationary points.
=⇒ Find the stationary point with the highest function value and you
find the solution.
Alexandra Schwartz (with J. Franke, C. Kanzow, and W. Leininger)
12/17
Two Examples
α
K (α)
x (α)
θ0 (α)
neutral weights
(0.25, 0.25, 0.25, 0.25)
{1, 2, 3}
(0.24, 0.08, 0.08, 0)
0.4
optimal weights
(0.143, 0.243, 0.243, 0.371)
{1, 2, 3, 4}
(0.238, 0.100, 0.100, 0.013)
0.451
Table: β = (1, 2, 2, 4)
α
K (α)
x (α)
θ0 (α)
neutral weigths
(0.25, 0.25, 0.25, 0.25)
{1, 2, 3}
(0.24, 0.08, 0.08, 0)
0.4
optimal weights
(0.226, 0.387, 0.387, 0)
{1, 2, 3}
(0.240, 0.105, 0.105, 0)
0.45
Table: β = (1, 2, 2, 6)
Alexandra Schwartz (with J. Franke, C. Kanzow, and W. Leininger)
13/17
Properties of the Solution
I
All global maxima share the same set of active players




X
K ∗ = i = 1, . . . , n (k ∗ − 2)βi <
βj


j∈K ∗
I
I
I
I
I
with k ∗ := |K ∗ |.
The strongest players, i.e. those with the smallest βi are active in
the solution.
At least three players are active, i.e. k ∗ ≥ 3.
By choosing the optimal weights, weaker participants are motivated
to participate actively.
Players with higher costs βi get bigger weights αi∗ .
Nonetheless, the effective cost parameter αβ∗i is still increasing in βi ,
i
i.e. the differences are not completely removed.
Alexandra Schwartz (with J. Franke, C. Kanzow, and W. Leininger)
14/17
More General Utility Functions
Instead of considering the utility functions
αi xi
θi (xi , x−i ; α) = PN
− βi xi ,
j=1 αj xj
sometimes the more general version
αi c(xi )
θi (xi , x−i ; α) = PN
− βi xi ,
j=1 αj c(xj )
with c(0) = 0 and c 0 (t) > 0 for all t ≥ 0 is of interest.
Existence and properties of solutions of the contest game?
Existence and properties of solutions of the effort maximization problem?
Alexandra Schwartz (with J. Franke, C. Kanzow, and W. Leininger)
15/17
Reformulation as MPCC
It is well known that (under certain conditions) Nash equilibrium
problems can be reformulated as variational inequalities. The NEP from
the last slide can be replaced by complementarity conditions if we
additionally assume c 00 (t) ≤ 0 for all t ≥ 0.
=⇒ Even if no explicit solution of the NEP is known, we could try to
solve the effort maximization problem numerically.
Properties of the reformulated problem?
Alexandra Schwartz (with J. Franke, C. Kanzow, and W. Leininger)
16/17
Conclusion
What did we gain?
=⇒ The colleagues from economy are happy.
=⇒ Additionally, we have a new – nontrivial – testproblem for
MPCC-solvers for which we know the solution.
Thank you very much for your
attention!
Alexandra Schwartz (with J. Franke, C. Kanzow, and W. Leininger)
17/17