Aspects of Waiting and Contracting in Game Theory

Doctoral thesis, 2015

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.