Here is a description of a modified producer -scrounger game. Pairs of shrews forage in a patch. A shrew can pursue either of two tactics: (1) search for food, called a producer or (2) take some of the food found by a producer, called a scrounger. Producing comes at a cost in terms of energy spent while searching for food), whereas scrounging is cost-free. A producer gets a finder's share (75% of the food) when paired with a scrounger (who gets 25% of the food). Producers share the search costs and the food. (40 points, 10 points per section) F= benefit, in terms of the amount of energy in the food C = cost, in terms of energy spent searching for food Payoff: Producer paired with a Producer = 12F-12C Payoff: Producer paired with a Scrounger = 3/4 F-C Payoff: Scrounger paired with a Producer = 1/4 F Payoff: Scrounger paired with a Scrounger = 0 1. Draw the payoff matrix for this game. 2. What is the ESS when F = 10 and C = 2? Make sure to calculate the equilibrium proportion of each tactic if the ESS is mixed. 3. The cost of search goes up, because food is harder to locate. What is the ESS when F = 10 and C=6? Make sure to calculate the equilibrium proportion of each tactic if the ESS is mixed. 4. In general, under what conditions is the ESS always pure Producer (i.e. no scroungers). Hint: When is it true that Payoff Producer against Producer > Payoff Scrounger against Producer and Payoff Producer against Producer > Payoff Scrounger against Scrounger ? Show you work to derive this. Here is a description of a modified producer -scrounger game. Pairs of shrews forage in a patch. A shrew can pursue either of two tactics: (1) search for food, called a producer or (2) take some of the food found by a producer, called a scrounger. Producing comes at a cost in terms of energy spent while searching for food), whereas scrounging is cost-free. A producer gets a finder's share (75% of the food) when paired with a scrounger (who gets 25% of the food). Producers share the search costs and the food. (40 points, 10 points per section) F= benefit, in terms of the amount of energy in the food C = cost, in terms of energy spent searching for food Payoff: Producer paired with a Producer = 12F-12C Payoff: Producer paired with a Scrounger = 3/4 F-C Payoff: Scrounger paired with a Producer = 1/4 F Payoff: Scrounger paired with a Scrounger = 0 1. Draw the payoff matrix for this game. 2. What is the ESS when F = 10 and C = 2? Make sure to calculate the equilibrium proportion of each tactic if the ESS is mixed. 3. The cost of search goes up, because food is harder to locate. What is the ESS when F = 10 and C=6? Make sure to calculate the equilibrium proportion of each tactic if the ESS is mixed. 4. In general, under what conditions is the ESS always pure Producer (i.e. no scroungers). Hint: When is it true that Payoff Producer against Producer > Payoff Scrounger against Producer and Payoff Producer against Producer > Payoff Scrounger against Scrounger ? Show you work to derive this