Aspects of Waiting and Contracting in Game Theory
The topic of this thesis concerns two selected problem in game theory; the Nplayer War of Attrition and the Principal-Agent problem.
The War of Attrition is a well established game theoretic model that was first
introduced in the 2-player case by John Maynard Smith. Although the original idea
was to describe certain animal behaviour in for example territorial competition the
model has also found interesting applications in economic theory. Following the
results of Maynard Smith, John Haigh and Chris Cannings generalised the War of
Attrition to two different models involving more than 2 players. We have chosen to
call these generalizations the dynamic- and the static model.
In Paper I we study the asymptotic behavior of the N-player models as the number
of players tend to infinity. By a thorough analysis of the dynamic model we find
a connection to the more difficult static model in the infinite regime. This connection
is then confirmed by approaching the limit of infinitely many players also in
the static model. By using these limit results as a source of inspiration for the finite
case, we manage to prove new results concerning existence and non-existence of
an equilibrium strategy in the N-player static case.
Principal-Agent problems constitute a class of models in economics treating optimal
incentives, often under moral hazard. A typical motivation is for instance
optimal contract formation between employer and employee. In 1987 Bengt Holmström
and Paul Milgrom wrote an influential paper on the subject, considering a
discrete time model. This work has served as one of the main references for more
recent studies on continuous time models.
Paper II and Paper III consider aspects of continuous time Principal-Agent problems
by a stochastic maximum principle approach, thus not relying on the dynamic
programming principle. In Paper II we introduce the Hidden Action model and the
Hidden Contract model and characterize optimal contracts in these by following a
scheme of sequential optimization. We also suggest a possible approach for solving
a strong formulation of the Hidden Actionmodel. Paper III generalizes the approach
of Paper II to involve models having time inconsistent utility functions. By doing so
we are able to consider Principal-Agent problems describing a risk averse behaviour,
for instance in the sense of mean-variance.
Time Inconsistent Utility Functions
Stochastic Maximum Principle
War of Attrition
Pontryagin’s Maximum Principle