This is a question of economic strategic analysis, which needs to be completed within 30 minutes

A creature has two actions when it nds a food source that another creature has found: battle the other creature to try and win the food for itself, or offer to split it. There are two and only two types of personalities in this large group of creatures. One personality always battles, and the other always splits. A food source is worth 200 kilojoules of energy and that energy determines each creature's fitness for reproduction. 'Battlers' are identical and so each has a 50% chance of winning a battle against another battler, but these ghts cost energy. If a battler meets a splitting type, the battler takes all of the food. If a splitting type meets another splitting type, they split the food without the cost of a battle. (a) Let the cost of a battle be 50 kilojoules (for each creature) and that when two battlers meet, each has a 50% chance to win and get all the food source's energy or to lose and get no energy. Draw the payoff table in terms of expected kilojoules for the game played between two random, risk-neutral creatures from the group. Find all evolutionary stable strategies (ESSs). (6 marks) (b) Now let the cost of a battle be 150 kilojoules for each battler. Draw the new payoff table and find all of the E885 for the group in this case. (6 marks) Now assume again that the cost of battling is 150 kilojoules. Imagine that a third personality type exists in the group of creatures. That type, the 'mixer', plays a mixed strategy whereby it sometimes battles and sometimes splits. (c) Find the mixed strategy equilibrium for the 2x2 game that is described by your payoff table from part b). (2 marks) (d) Draw the 3x3 payoff table for this game with the mixer type, assuming that this type uses the mixed strategy that you found in part c). (2 marks) (e) Prove whether or not there is an ESS of this game in which the group is made up entirely of the mixing type. (Hint: Test whether a group of mixers can be successfully invaded by either of the other two types.) (4 marks)