## Essays in game theory and bankruptcy

##### Abstract

In Chapter 1 I study the iterative strategy elimination mechanisms for normal
form games. The literature is mostly clustered around the order of elimination.
The conventional elimination also requires more strict knowledge assumptions if
the elimination is iterative. I define an elimination process which requires weaker
rationality. I establish some preliminary results suggesting that my mechanism is
order independent whenever iterative elimination of weakly dominated strategies
(IEWDS) is so. I also specify conditions under which the \undercutting problem"
occurs. Comparison of other elimination mechanisms in the literature (Iterated
Weak Strategy Elimination, Iterated Strict Strategy Elimination, Generalized
Strategy Eliminability Criterion, RBEU, Dekel-Fudenberg Procedure, Asheim-
Dufwenberg Procedure) and mine is also studied to some extent. In Chapter 2
I study the axiomatic characterization of a well-known bankruptcy rule: Proportional
Division (PROP). The rule allocates shares proportional to agents'
claims and hence, is intuitive according to many authors. I give supporting
evidence to this opinion by first defining a new type of consistency requirement,
i.e. union-consistency and showing that PROP is the only rule that satisfies
anonymity, continuity and union-consistency. Note that anonymity and continuity
are very general requirements and satisfied by almost all the rules that have
been studied in this literature. Thus, I prove that we can choose a unique rule
among them by only requiring union-consistency. Then, I define a bankruptcy
operator and give some intuition on it. A bankruptcy operator is a mapping from
the set of bankruptcy operators to itself. I prove that any rule will converge to
PROP under this operator as the claims increase. I show nice characteristics of
the operator some of which are related to PROP. I also give a definition for continuity
of an operator. In Chapter 3 investigate risk-averse investors' behaviour
towards a risky firm. In order to find Pareto Optimal allocations regarding a joint
venture, I employ a 2-stage game, first stage of which involves a social-planner
committing to an ex-post bankruptcy rule. A bankruptcy rule is a set of suggestions
for solving each possible bankruptcy problem. A bankruptcy problem
occurs when there is not enough endowment to allocate to the agents each of
whom has a claim on it. I devise the game-theoretic approach posed in K1br1s
and K1br1s (2013) and extend it further. In fact, that paper considers a comparison
among 4 renowned bankruptcy rules whereas mine do not restrict attention
to any particular rule but rather aim to find a Pareto Optimal(PO) one. I start
with 2 agent case in order to give some insight to the reader and then, generalise
the results to an arbitrary number of investors. I find socially desirable (PO)
allocations and show that the same can be achieved through financial markets by
the help of some well-known results.