Enumerate the following problems and provide answers.

1 Consider the following game in strategic form. I L U S M N 2,10 4,9 4,8 4,5 2,11 Y 9,4 3,8 4,2 5,5 8,1 Define the concept of a strictly dominated strategy as one which gives a strictly smaller payoff than another strategy for whichever strategy any other opponent plays. In contrast, a weakly dominated strategy is one which gives a payoff that is at most as good as that given by another strategy for whichever strategy any other opponent plays. If after elimination of one weakly or strictly dominated strategy at least one other strategy becomes weakly/strictly dominated, then we can, in an iterative process, delete also this/these latter strategy/ies. In the case of eliminating weakly dominated strategies, the order of elimination matters in a sense that a different subset of strategies may survive the process of elimination, because strategies might have been eliminated that were as good as another given at least one strategy of the opponent(s). a) Show whether the above game is dominance solvable by identifying any strictly dominated strategies and, if there are any, show for which probability ranges which strategies are strictly dominated. [30%] b) Define the concept of a Nash equilibrium and find all Nash equilibria in pure and mixed strategies in the provided game. [40%] c) Find the unique Nash equilibrium in mixed strategies of the game below. [30%] R T V W 5,9 7,8 7,7 X 6,8 8,7 5,9 Z 7,7 6,9 6,8 Please define every concept carefully before you use it....

At the end of each year an independent organisation ranks the performance of the unit trusts invested in a particular sector, and classifies them into four quartiles (with quartile 1 being the best performer). Past experience has shown that, at the end of each year, a fund will either remain in the same quartile or will move to a neighbouring quartile. In fact, there is a probability 1-2c that a fund will remain in the same quartile and, where upward or downward movements are both possible, these are equally likely. However, it has been found that a fund that has remained in the top or bottom quartile for two consecutive years has a probability of 1-B (B